Theorem onsucuni2 3081

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

Assertion

Ref	Expression
onsucuni2	|- ((A e. On /\ A = suc B) -> suc U.A = A)

Proof of Theorem onsucuni2

Step	Hyp	Ref	Expression
1		eleq1 1526	. . . . . 6 |- (A = suc B -> (A e. On <-> suc B e. On))
2		sucelon 3058	. . . . . 6 |- (B e. On <-> suc B e. On)
3	1, 2	syl6bbr 536	. . . . 5 |- (A = suc B -> (A e. On <-> B e. On))
4	3	biimpac 418	. . . 4 |- ((A e. On /\ A = suc B) -> B e. On)
5		eloni 2948	. . . . . . . . . . 11 |- (B e. On -> Ord B)
6		ordirr 2956	. . . . . . . . . . 11 |- (Ord B -> -. B e. B)
7	5, 6	syl 10	. . . . . . . . . 10 |- (B e. On -> -. B e. B)
8		eleq2 1527	. . . . . . . . . . 11 |- (suc B = B -> (B e. suc B <-> B e. B))
9		sucidg 3042	. . . . . . . . . . 11 |- (B e. On -> B e. suc B)
10	8, 9	syl5cbi 209	. . . . . . . . . 10 |- (B e. On -> (suc B = B -> B e. B))
11	7, 10	mtod 108	. . . . . . . . 9 |- (B e. On -> -. suc B = B)
12		ordunisuc 3079	. . . . . . . . . . 11 |- (Ord B -> U.suc B = B)
13	5, 12	syl 10	. . . . . . . . . 10 |- (B e. On -> U.suc B = B)
14	13	eqeq2d 1478	. . . . . . . . 9 |- (B e. On -> (suc B = U.suc B <-> suc B = B))
15	11, 14	mtbird 713	. . . . . . . 8 |- (B e. On -> -. suc B = U.suc B)
16	15	adantl 388	. . . . . . 7 |- ((A = suc B /\ B e. On) -> -. suc B = U.suc B)
17		id 59	. . . . . . . . 9 |- (A = suc B -> A = suc B)
18		unieq 2500	. . . . . . . . 9 |- (A = suc B -> U.A = U.suc B)
19	17, 18	eqeq12d 1481	. . . . . . . 8 |- (A = suc B -> (A = U.A <-> suc B = U.suc B))
20	19	adantr 389	. . . . . . 7 |- ((A = suc B /\ B e. On) -> (A = U.A <-> suc B = U.suc B))
21	16, 20	mtbird 713	. . . . . 6 |- ((A = suc B /\ B e. On) -> -. A = U.A)
22	21	ex 373	. . . . 5 |- (A = suc B -> (B e. On -> -. A = U.A))
23	22	adantl 388	. . . 4 |- ((A e. On /\ A = suc B) -> (B e. On -> -. A = U.A))
24	4, 23	mpd 26	. . 3 |- ((A e. On /\ A = suc B) -> -. A = U.A)
25		eloni 2948	. . . . 5 |- (A e. On -> Ord A)
26		orduniorsuc 3077	. . . . . 6 |- (Ord A -> (A = U.A \/ A = suc U.A))
27	26	ord 232	. . . . 5 |- (Ord A -> (-. A = U.A -> A = suc U.A))
28	25, 27	syl 10	. . . 4 |- (A e. On -> (-. A = U.A -> A = suc U.A))
29	28	adantr 389	. . 3 |- ((A e. On /\ A = suc B) -> (-. A = U.A -> A = suc U.A))
30	24, 29	mpd 26	. 2 |- ((A e. On /\ A = suc B) -> A = suc U.A)
31	30	eqcomd 1472	1 |- ((A e. On /\ A = suc B) -> suc U.A = A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  U.cuni 2493  Ord word 2937  Oncon0 2938  suc csuc 2940

This theorem is referenced by:  rankxplim3 4686  rankxpsuc 4687

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-suc 2944

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