Theorem oa00 3161

Description: An ordinal sum is zero iff both of its arguments are zero.

Assertion

Ref	Expression
oa00	⊢ ((A ∈ On ∧ B ∈ On) → ((A +o B) = ∅ ↔ (A = ∅ ∧ B = ∅)))

Proof of Theorem oa00

Step	Hyp	Ref	Expression
1		on0eln0 2279	. . . . . . 7 ⊢ (A ∈ On → (∅ ∈ A ↔ ¬ A = ∅))
2	1	adantr 306	. . . . . 6 ⊢ ((A ∈ On ∧ B ∈ On) → (∅ ∈ A ↔ ¬ A = ∅))
3		oaword1 3154	. . . . . . 7 ⊢ ((A ∈ On ∧ B ∈ On) → A ⊆ (A +o B))
4	3	sseld 1506	. . . . . 6 ⊢ ((A ∈ On ∧ B ∈ On) → (∅ ∈ A → ∅ ∈ (A +o B)))
5	2, 4	sylbird 180	. . . . 5 ⊢ ((A ∈ On ∧ B ∈ On) → (¬ A = ∅ → ∅ ∈ (A +o B)))
6		n0i 1712	. . . . 5 ⊢ (∅ ∈ (A +o B) → ¬ (A +o B) = ∅)
7	5, 6	syl6 23	. . . 4 ⊢ ((A ∈ On ∧ B ∈ On) → (¬ A = ∅ → ¬ (A +o B) = ∅))
8	7	a3d 70	. . 3 ⊢ ((A ∈ On ∧ B ∈ On) → ((A +o B) = ∅ → A = ∅))
9		on0eln0 2279	. . . . . . 7 ⊢ (B ∈ On → (∅ ∈ B ↔ ¬ B = ∅))
10	9	adantl 305	. . . . . 6 ⊢ ((A ∈ On ∧ B ∈ On) → (∅ ∈ B ↔ ¬ B = ∅))
11		0elon 2277	. . . . . . . 8 ⊢ ∅ ∈ On
12		oaord 3149	. . . . . . . 8 ⊢ ((∅ ∈ On ∧ B ∈ On ∧ A ∈ On) → (∅ ∈ B ↔ (A +o ∅) ∈ (A +o B)))
13	11, 12	mp3an1 639	. . . . . . 7 ⊢ ((B ∈ On ∧ A ∈ On) → (∅ ∈ B ↔ (A +o ∅) ∈ (A +o B)))
14	13	ancoms 334	. . . . . 6 ⊢ ((A ∈ On ∧ B ∈ On) → (∅ ∈ B ↔ (A +o ∅) ∈ (A +o B)))
15	10, 14	bitr3d 408	. . . . 5 ⊢ ((A ∈ On ∧ B ∈ On) → (¬ B = ∅ ↔ (A +o ∅) ∈ (A +o B)))
16		n0i 1712	. . . . 5 ⊢ ((A +o ∅) ∈ (A +o B) → ¬ (A +o B) = ∅)
17	15, 16	syl6bi 187	. . . 4 ⊢ ((A ∈ On ∧ B ∈ On) → (¬ B = ∅ → ¬ (A +o B) = ∅))
18	17	a3d 70	. . 3 ⊢ ((A ∈ On ∧ B ∈ On) → ((A +o B) = ∅ → B = ∅))
19	8, 18	jcad 455	. 2 ⊢ ((A ∈ On ∧ B ∈ On) → ((A +o B) = ∅ → (A = ∅ ∧ B = ∅)))
20		opreq12 3008	. . . 4 ⊢ ((A = ∅ ∧ B = ∅) → (A +o B) = (∅ +o ∅))
21		oa0 3124	. . . . 5 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅)
22	11, 21	ax-mp 6	. . . 4 ⊢ (∅ +o ∅) = ∅
23	20, 22	syl6eq 1140	. . 3 ⊢ ((A = ∅ ∧ B = ∅) → (A +o B) = ∅)
24	23	a1i 7	. 2 ⊢ ((A ∈ On ∧ B ∈ On) → ((A = ∅ ∧ B = ∅) → (A +o B) = ∅))
25	19, 24	impbid 397	1 ⊢ ((A ∈ On ∧ B ∈ On) → ((A +o B) = ∅ ↔ (A = ∅ ∧ B = ∅)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∅c0 1707  Oncon0 2199  (class class class)co 3001   +_o coa 3101

This theorem is referenced by:  oalimcl 3162

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-oadd 3106

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