Theorem oen0 3165

Description: Ordinal exponentiation with a nonzero mantissa is nonzero. Proposition 8.32 of [TakeutiZaring] p. 67.

Assertion

Ref	Expression
oen0	⊢ (((A ∈ On ∧ B ∈ On) ∧ ∅ ∈ A) → ∅ ∈ (A ↑o B))

Proof of Theorem oen0

Step	Hyp	Ref	Expression
1		opreq2 3007	. . . . . 6 ⊢ (x = ∅ → (A ↑o x) = (A ↑o ∅))
2	1	eleq2d 1156	. . . . 5 ⊢ (x = ∅ → (∅ ∈ (A ↑o x) ↔ ∅ ∈ (A ↑o ∅)))
3		opreq2 3007	. . . . . 6 ⊢ (x = y → (A ↑o x) = (A ↑o y))
4	3	eleq2d 1156	. . . . 5 ⊢ (x = y → (∅ ∈ (A ↑o x) ↔ ∅ ∈ (A ↑o y)))
5		opreq2 3007	. . . . . 6 ⊢ (x = suc y → (A ↑o x) = (A ↑o suc y))
6	5	eleq2d 1156	. . . . 5 ⊢ (x = suc y → (∅ ∈ (A ↑o x) ↔ ∅ ∈ (A ↑o suc y)))
7		opreq2 3007	. . . . . 6 ⊢ (x = B → (A ↑o x) = (A ↑o B))
8	7	eleq2d 1156	. . . . 5 ⊢ (x = B → (∅ ∈ (A ↑o x) ↔ ∅ ∈ (A ↑o B)))
9		0lt1o 3116	. . . . . . 7 ⊢ ∅ ∈ 1o
10		oe0 3130	. . . . . . . 8 ⊢ (A ∈ On → (A ↑o ∅) = 1o)
11	10	eleq2d 1156	. . . . . . 7 ⊢ (A ∈ On → (∅ ∈ (A ↑o ∅) ↔ ∅ ∈ 1o))
12	9, 11	mpbiri 169	. . . . . 6 ⊢ (A ∈ On → ∅ ∈ (A ↑o ∅))
13	12	adantr 306	. . . . 5 ⊢ ((A ∈ On ∧ ∅ ∈ A) → ∅ ∈ (A ↑o ∅))
14		omordi 3164	. . . . . . . . . . . 12 ⊢ (((A ∈ On ∧ (A ↑o y) ∈ On) ∧ ∅ ∈ (A ↑o y)) → (∅ ∈ A → ((A ↑o y) ·o ∅) ∈ ((A ↑o y) ·o A)))
15		om0 3125	. . . . . . . . . . . . . 14 ⊢ ((A ↑o y) ∈ On → ((A ↑o y) ·o ∅) = ∅)
16	15	eleq1d 1155	. . . . . . . . . . . . 13 ⊢ ((A ↑o y) ∈ On → (((A ↑o y) ·o ∅) ∈ ((A ↑o y) ·o A) ↔ ∅ ∈ ((A ↑o y) ·o A)))
17	16	ad2antlr 321	. . . . . . . . . . . 12 ⊢ (((A ∈ On ∧ (A ↑o y) ∈ On) ∧ ∅ ∈ (A ↑o y)) → (((A ↑o y) ·o ∅) ∈ ((A ↑o y) ·o A) ↔ ∅ ∈ ((A ↑o y) ·o A)))
18	14, 17	sylibd 177	. . . . . . . . . . 11 ⊢ (((A ∈ On ∧ (A ↑o y) ∈ On) ∧ ∅ ∈ (A ↑o y)) → (∅ ∈ A → ∅ ∈ ((A ↑o y) ·o A)))
19		pm3.26 256	. . . . . . . . . . . 12 ⊢ ((A ∈ On ∧ y ∈ On) → A ∈ On)
20		oecl 3140	. . . . . . . . . . . 12 ⊢ ((A ∈ On ∧ y ∈ On) → (A ↑o y) ∈ On)
21	19, 20	jca 236	. . . . . . . . . . 11 ⊢ ((A ∈ On ∧ y ∈ On) → (A ∈ On ∧ (A ↑o y) ∈ On))
22	18, 21	sylan 343	. . . . . . . . . 10 ⊢ (((A ∈ On ∧ y ∈ On) ∧ ∅ ∈ (A ↑o y)) → (∅ ∈ A → ∅ ∈ ((A ↑o y) ·o A)))
23		oesuc 3134	. . . . . . . . . . . 12 ⊢ ((A ∈ On ∧ y ∈ On) → (A ↑o suc y) = ((A ↑o y) ·o A))
24	23	eleq2d 1156	. . . . . . . . . . 11 ⊢ ((A ∈ On ∧ y ∈ On) → (∅ ∈ (A ↑o suc y) ↔ ∅ ∈ ((A ↑o y) ·o A)))
25	24	adantr 306	. . . . . . . . . 10 ⊢ (((A ∈ On ∧ y ∈ On) ∧ ∅ ∈ (A ↑o y)) → (∅ ∈ (A ↑o suc y) ↔ ∅ ∈ ((A ↑o y) ·o A)))
26	22, 25	sylibrd 179	. . . . . . . . 9 ⊢ (((A ∈ On ∧ y ∈ On) ∧ ∅ ∈ (A ↑o y)) → (∅ ∈ A → ∅ ∈ (A ↑o suc y)))
27	26	exp31 293	. . . . . . . 8 ⊢ (A ∈ On → (y ∈ On → (∅ ∈ (A ↑o y) → (∅ ∈ A → ∅ ∈ (A ↑o suc y)))))
28	27	com12 13	. . . . . . 7 ⊢ (y ∈ On → (A ∈ On → (∅ ∈ (A ↑o y) → (∅ ∈ A → ∅ ∈ (A ↑o suc y)))))
29	28	com34 36	. . . . . 6 ⊢ (y ∈ On → (A ∈ On → (∅ ∈ A → (∅ ∈ (A ↑o y) → ∅ ∈ (A ↑o suc y)))))
30	29	imp3a 279	. . . . 5 ⊢ (y ∈ On → ((A ∈ On ∧ ∅ ∈ A) → (∅ ∈ (A ↑o y) → ∅ ∈ (A ↑o suc y))))
31		0ellim 2285	. . . . . . . . . . . 12 ⊢ (Lim x → ∅ ∈ x)
32		eqimss2 1549	. . . . . . . . . . . . 13 ⊢ ((A ↑o ∅) = 1o → 1o ⊆ (A ↑o ∅))
33	10, 32	syl 12	. . . . . . . . . . . 12 ⊢ (A ∈ On → 1o ⊆ (A ↑o ∅))
34	31, 33	anim12i 268	. . . . . . . . . . 11 ⊢ ((Lim x ∧ A ∈ On) → (∅ ∈ x ∧ 1o ⊆ (A ↑o ∅)))
35		opreq2 3007	. . . . . . . . . . . . 13 ⊢ (y = ∅ → (A ↑o y) = (A ↑o ∅))
36	35	sseq2d 1528	. . . . . . . . . . . 12 ⊢ (y = ∅ → (1o ⊆ (A ↑o y) ↔ 1o ⊆ (A ↑o ∅)))
37	36	rcla4ev 1403	. . . . . . . . . . 11 ⊢ ((∅ ∈ x ∧ 1o ⊆ (A ↑o ∅)) → ∃y ∈ x 1o ⊆ (A ↑o y))
38		ssiun 2018	. . . . . . . . . . 11 ⊢ (∃y ∈ x 1o ⊆ (A ↑o y) → 1o ⊆ ∪y ∈ x (A ↑o y))
39	34, 37, 38	3syl 21	. . . . . . . . . 10 ⊢ ((Lim x ∧ A ∈ On) → 1o ⊆ ∪y ∈ x (A ↑o y))
40	39	adantrr 312	. . . . . . . . 9 ⊢ ((Lim x ∧ (A ∈ On ∧ ∅ ∈ A)) → 1o ⊆ ∪y ∈ x (A ↑o y))
41		visset 1350	. . . . . . . . . . . 12 ⊢ x ∈ V
42		oelim 3137	. . . . . . . . . . . 12 ⊢ (((A ∈ On ∧ (x ∈ V ∧ Lim x)) ∧ ∅ ∈ A) → (A ↑o x) = ∪y ∈ x (A ↑o y))
43	41, 42	mpan121 533	. . . . . . . . . . 11 ⊢ (((A ∈ On ∧ Lim x) ∧ ∅ ∈ A) → (A ↑o x) = ∪y ∈ x (A ↑o y))
44	43	anasss 337	. . . . . . . . . 10 ⊢ ((A ∈ On ∧ (Lim x ∧ ∅ ∈ A)) → (A ↑o x) = ∪y ∈ x (A ↑o y))
45	44	an1s 372	. . . . . . . . 9 ⊢ ((Lim x ∧ (A ∈ On ∧ ∅ ∈ A)) → (A ↑o x) = ∪y ∈ x (A ↑o y))
46	40, 45	sseqtr4d 1537	. . . . . . . 8 ⊢ ((Lim x ∧ (A ∈ On ∧ ∅ ∈ A)) → 1o ⊆ (A ↑o x))
47		oecl 3140	. . . . . . . . . . . 12 ⊢ ((A ∈ On ∧ x ∈ On) → (A ↑o x) ∈ On)
48	47	ancoms 334	. . . . . . . . . . 11 ⊢ ((x ∈ On ∧ A ∈ On) → (A ↑o x) ∈ On)
49		limelon 2286	. . . . . . . . . . . 12 ⊢ ((x ∈ V ∧ Lim x) → x ∈ On)
50	41, 49	mpan 518	. . . . . . . . . . 11 ⊢ (Lim x → x ∈ On)
51	48, 50	sylan 343	. . . . . . . . . 10 ⊢ ((Lim x ∧ A ∈ On) → (A ↑o x) ∈ On)
52		eloni 2209	. . . . . . . . . 10 ⊢ ((A ↑o x) ∈ On → Ord (A ↑o x))
53		ordgt0ge1 3114	. . . . . . . . . 10 ⊢ (Ord (A ↑o x) → (∅ ∈ (A ↑o x) ↔ 1o ⊆ (A ↑o x)))
54	51, 52, 53	3syl 21	. . . . . . . . 9 ⊢ ((Lim x ∧ A ∈ On) → (∅ ∈ (A ↑o x) ↔ 1o ⊆ (A ↑o x)))
55	54	adantrr 312	. . . . . . . 8 ⊢ ((Lim x ∧ (A ∈ On ∧ ∅ ∈ A)) → (∅ ∈ (A ↑o x) ↔ 1o ⊆ (A ↑o x)))
56	46, 55	mpbird 171	. . . . . . 7 ⊢ ((Lim x ∧ (A ∈ On ∧ ∅ ∈ A)) → ∅ ∈ (A ↑o x))
57	56	exp 291	. . . . . 6 ⊢ (Lim x → ((A ∈ On ∧ ∅ ∈ A) → ∅ ∈ (A ↑o x)))
58	57	a1dd 42	. . . . 5 ⊢ (Lim x → ((A ∈ On ∧ ∅ ∈ A) → (∀y ∈ x ∅ ∈ (A ↑o y) → ∅ ∈ (A ↑o x))))
59	2, 4, 6, 8, 13, 30, 58	tfinds3 2406	. . . 4 ⊢ (B ∈ On → ((A ∈ On ∧ ∅ ∈ A) → ∅ ∈ (A ↑o B)))
60	59	exp3a 292	. . 3 ⊢ (B ∈ On → (A ∈ On → (∅ ∈ A → ∅ ∈ (A ↑o B))))
61	60	com12 13	. 2 ⊢ (A ∈ On → (B ∈ On → (∅ ∈ A → ∅ ∈ (A ↑o B))))
62	61	imp31 280	1 ⊢ (((A ∈ On ∧ B ∈ On) ∧ ∅ ∈ A) → ∅ ∈ (A ↑o B))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  ∪ciun 1994  Ord word 2198  Oncon0 2199  Lim wlim 2200  suc csuc 2201  (class class class)co 3001  1_oc1o 3099   ·_o comu 3102   ↑_o coe 3103

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-oexp 3108

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