Theorem onsucuni2 3091

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 1534	. . . . . 6 |- (A = suc B -> (A e. On <-> suc B e. On))
2		sucelon 3068	. . . . . 6 |- (B e. On <-> suc B e. On)
3	1, 2	syl6bbr 538	. . . . 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 2958	. . . . . . . . . . 11 |- (B e. On -> Ord B)
6		ordirr 2966	. . . . . . . . . . 11 |- (Ord B -> -. B e. B)
7	5, 6	syl 10	. . . . . . . . . 10 |- (B e. On -> -. B e. B)
8		eleq2 1535	. . . . . . . . . . 11 |- (suc B = B -> (B e. suc B <-> B e. B))
9		sucidg 3052	. . . . . . . . . . 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 3089	. . . . . . . . . . 11 |- (Ord B -> U.suc B = B)
13	5, 12	syl 10	. . . . . . . . . 10 |- (B e. On -> U.suc B = B)
14	13	eqeq2d 1486	. . . . . . . . 9 |- (B e. On -> (suc B = U.suc B <-> suc B = B))
15	11, 14	mtbird 715	. . . . . . . 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 2510	. . . . . . . . 9 |- (A = suc B -> U.A = U.suc B)
19	17, 18	eqeq12d 1489	. . . . . . . 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 715	. . . . . 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 2958	. . . . 5 |- (A e. On -> Ord A)
26		orduniorsuc 3087	. . . . . 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 1480	1 |- ((A e. On /\ A = suc B) -> suc U.A = A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  U.cuni 2503  Ord word 2947  Oncon0 2948  suc csuc 2950

This theorem is referenced by:  rankxplim3 4714  rankxpsuc 4715

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954

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