Theorem oaordex 3160

Description: Existence theorem for ordering of ordinal sum. Similar to Proposition 4.34(f) of [Mendelson] p. 266 and its converse.

Assertion

Ref	Expression
oaordex	⊢ ((A ∈ On ∧ B ∈ On) → (A ∈ B ↔ ∃x ∈ On (∅ ∈ x ∧ (A +o x) = B)))

Distinct variable group(s):   x,A   x,B

Proof of Theorem oaordex

Step	Hyp	Ref	Expression
1		onelsst 2255	. . . . 5 ⊢ (B ∈ On → (A ∈ B → A ⊆ B))
2	1	adantl 305	. . . 4 ⊢ ((A ∈ On ∧ B ∈ On) → (A ∈ B → A ⊆ B))
3		oawordex 3159	. . . 4 ⊢ ((A ∈ On ∧ B ∈ On) → (A ⊆ B ↔ ∃x ∈ On (A +o x) = B))
4	2, 3	sylibd 177	. . 3 ⊢ ((A ∈ On ∧ B ∈ On) → (A ∈ B → ∃x ∈ On (A +o x) = B))
5		oaord1 3153	. . . . . . . . . . . . 13 ⊢ ((A ∈ On ∧ x ∈ On) → (∅ ∈ x ↔ A ∈ (A +o x)))
6		eleq2 1150	. . . . . . . . . . . . 13 ⊢ ((A +o x) = B → (A ∈ (A +o x) ↔ A ∈ B))
7	5, 6	sylan9bb 418	. . . . . . . . . . . 12 ⊢ (((A ∈ On ∧ x ∈ On) ∧ (A +o x) = B) → (∅ ∈ x ↔ A ∈ B))
8	7	biimprcd 138	. . . . . . . . . . 11 ⊢ (A ∈ B → (((A ∈ On ∧ x ∈ On) ∧ (A +o x) = B) → ∅ ∈ x))
9	8	exp4c 297	. . . . . . . . . 10 ⊢ (A ∈ B → (A ∈ On → (x ∈ On → ((A +o x) = B → ∅ ∈ x))))
10	9	com12 13	. . . . . . . . 9 ⊢ (A ∈ On → (A ∈ B → (x ∈ On → ((A +o x) = B → ∅ ∈ x))))
11	10	imp4b 283	. . . . . . . 8 ⊢ ((A ∈ On ∧ A ∈ B) → ((x ∈ On ∧ (A +o x) = B) → ∅ ∈ x))
12		pm3.27 260	. . . . . . . . 9 ⊢ ((x ∈ On ∧ (A +o x) = B) → (A +o x) = B)
13	12	a1i 7	. . . . . . . 8 ⊢ ((A ∈ On ∧ A ∈ B) → ((x ∈ On ∧ (A +o x) = B) → (A +o x) = B))
14	11, 13	jcad 455	. . . . . . 7 ⊢ ((A ∈ On ∧ A ∈ B) → ((x ∈ On ∧ (A +o x) = B) → (∅ ∈ x ∧ (A +o x) = B)))
15	14	exp3a 292	. . . . . 6 ⊢ ((A ∈ On ∧ A ∈ B) → (x ∈ On → ((A +o x) = B → (∅ ∈ x ∧ (A +o x) = B))))
16	15	r19.22dv 1278	. . . . 5 ⊢ ((A ∈ On ∧ A ∈ B) → (∃x ∈ On (A +o x) = B → ∃x ∈ On (∅ ∈ x ∧ (A +o x) = B)))
17	16	exp 291	. . . 4 ⊢ (A ∈ On → (A ∈ B → (∃x ∈ On (A +o x) = B → ∃x ∈ On (∅ ∈ x ∧ (A +o x) = B))))
18	17	adantr 306	. . 3 ⊢ ((A ∈ On ∧ B ∈ On) → (A ∈ B → (∃x ∈ On (A +o x) = B → ∃x ∈ On (∅ ∈ x ∧ (A +o x) = B))))
19	4, 18	mpdd 47	. 2 ⊢ ((A ∈ On ∧ B ∈ On) → (A ∈ B → ∃x ∈ On (∅ ∈ x ∧ (A +o x) = B)))
20	7	biimpd 135	. . . . . . 7 ⊢ (((A ∈ On ∧ x ∈ On) ∧ (A +o x) = B) → (∅ ∈ x → A ∈ B))
21	20	exp31 293	. . . . . 6 ⊢ (A ∈ On → (x ∈ On → ((A +o x) = B → (∅ ∈ x → A ∈ B))))
22	21	com34 36	. . . . 5 ⊢ (A ∈ On → (x ∈ On → (∅ ∈ x → ((A +o x) = B → A ∈ B))))
23	22	imp4a 282	. . . 4 ⊢ (A ∈ On → (x ∈ On → ((∅ ∈ x ∧ (A +o x) = B) → A ∈ B)))
24	23	r19.23adv 1286	. . 3 ⊢ (A ∈ On → (∃x ∈ On (∅ ∈ x ∧ (A +o x) = B) → A ∈ B))
25	24	adantr 306	. 2 ⊢ ((A ∈ On ∧ B ∈ On) → (∃x ∈ On (∅ ∈ x ∧ (A +o x) = B) → A ∈ B))
26	19, 25	impbid 397	1 ⊢ ((A ∈ On ∧ B ∈ On) → (A ∈ B ↔ ∃x ∈ On (∅ ∈ x ∧ (A +o x) = B)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∃wrex 1202   ⊆ wss 1487  ∅c0 1707  Oncon0 2199  (class class class)co 3001   +_o coa 3101

This theorem is referenced by:  nnaordex 3191

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-reu 1207  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-int 1966  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-oadd 3106

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