Theorem onsucuni2OLD 3915

Description: A successor ordinal is the successor of its union.

Assertion

Ref	Expression
onsucuni2OLD	|- ((A e. On /\ A = suc B) -> suc U.A = A)

Proof of Theorem onsucuni2OLD

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . . . 6 |- (A = suc B -> (A e. On <-> suc B e. On))
2		sucelon 3898	. . . . . 6 |- (B e. On <-> suc B e. On)
3	1, 2	syl6bbr 597	. . . . 5 |- (A = suc B -> (A e. On <-> B e. On))
4	3	biimpac 462	. . . 4 |- ((A e. On /\ A = suc B) -> B e. On)
5		eloni 3667	. . . . . . . . . . 11 |- (B e. On -> Ord B)
6		ordirr 3676	. . . . . . . . . . 11 |- (Ord B -> -. B e. B)
7	5, 6	syl 12	. . . . . . . . . 10 |- (B e. On -> -. B e. B)
8		eleq2 1958	. . . . . . . . . . 11 |- (suc B = B -> (B e. suc B <-> B e. B))
9		sucidg 3743	. . . . . . . . . . 11 |- (B e. On -> B e. suc B)
10	8, 9	syl5cbi 226	. . . . . . . . . 10 |- (B e. On -> (suc B = B -> B e. B))
11	7, 10	mtod 123	. . . . . . . . 9 |- (B e. On -> -. suc B = B)
12		ordunisuc 3911	. . . . . . . . . . 11 |- (Ord B -> U.suc B = B)
13	5, 12	syl 12	. . . . . . . . . 10 |- (B e. On -> U.suc B = B)
14	13	eqeq2d 1895	. . . . . . . . 9 |- (B e. On -> (suc B = U.suc B <-> suc B = B))
15	11, 14	mtbird 783	. . . . . . . 8 |- (B e. On -> -. suc B = U.suc B)
16	15	adantl 424	. . . . . . 7 |- ((A = suc B /\ B e. On) -> -. suc B = U.suc B)
17		id 73	. . . . . . . . 9 |- (A = suc B -> A = suc B)
18		unieq 3185	. . . . . . . . 9 |- (A = suc B -> U.A = U.suc B)
19	17, 18	eqeq12d 1899	. . . . . . . 8 |- (A = suc B -> (A = U.A <-> suc B = U.suc B))
20	19	adantr 425	. . . . . . 7 |- ((A = suc B /\ B e. On) -> (A = U.A <-> suc B = U.suc B))
21	16, 20	mtbird 783	. . . . . 6 |- ((A = suc B /\ B e. On) -> -. A = U.A)
22	21	ex 402	. . . . 5 |- (A = suc B -> (B e. On -> -. A = U.A))
23	22	adantl 424	. . . 4 |- ((A e. On /\ A = suc B) -> (B e. On -> -. A = U.A))
24	4, 23	mpd 29	. . 3 |- ((A e. On /\ A = suc B) -> -. A = U.A)
25		eloni 3667	. . . . 5 |- (A e. On -> Ord A)
26		orduniorsuc 3909	. . . . . 6 |- (Ord A -> (A = U.A \/ A = suc U.A))
27	26	ord 249	. . . . 5 |- (Ord A -> (-. A = U.A -> A = suc U.A))
28	25, 27	syl 12	. . . 4 |- (A e. On -> (-. A = U.A -> A = suc U.A))
29	28	adantr 425	. . 3 |- ((A e. On /\ A = suc B) -> (-. A = U.A -> A = suc U.A))
30	24, 29	mpd 29	. 2 |- ((A e. On /\ A = suc B) -> A = suc U.A)
31	30	eqcomd 1889	1 |- ((A e. On /\ A = suc B) -> suc U.A = A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  U.cuni 3177  Ord word 3656  Oncon0 3657  suc csuc 3659

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663

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