Theorem onsucuni2 3097

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 1537	. . . . . 6 |- (A = suc B -> (A e. On <-> suc B e. On))
2		sucelon 3074	. . . . . 6 |- (B e. On <-> suc B e. On)
3	1, 2	syl6bbr 540	. . . . 5 |- (A = suc B -> (A e. On <-> B e. On))
4	3	biimpac 420	. . . 4 |- ((A e. On /\ A = suc B) -> B e. On)
5		eloni 2964	. . . . . . . . . . 11 |- (B e. On -> Ord B)
6		ordirr 2972	. . . . . . . . . . 11 |- (Ord B -> -. B e. B)
7	5, 6	syl 10	. . . . . . . . . 10 |- (B e. On -> -. B e. B)
8		eleq2 1538	. . . . . . . . . . 11 |- (suc B = B -> (B e. suc B <-> B e. B))
9		sucidg 3058	. . . . . . . . . . 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 3095	. . . . . . . . . . 11 |- (Ord B -> U.suc B = B)
13	5, 12	syl 10	. . . . . . . . . 10 |- (B e. On -> U.suc B = B)
14	13	eqeq2d 1489	. . . . . . . . 9 |- (B e. On -> (suc B = U.suc B <-> suc B = B))
15	11, 14	mtbird 717	. . . . . . . 8 |- (B e. On -> -. suc B = U.suc B)
16	15	adantl 390	. . . . . . 7 |- ((A = suc B /\ B e. On) -> -. suc B = U.suc B)
17		id 59	. . . . . . . . 9 |- (A = suc B -> A = suc B)
18		unieq 2514	. . . . . . . . 9 |- (A = suc B -> U.A = U.suc B)
19	17, 18	eqeq12d 1492	. . . . . . . 8 |- (A = suc B -> (A = U.A <-> suc B = U.suc B))
20	19	adantr 391	. . . . . . 7 |- ((A = suc B /\ B e. On) -> (A = U.A <-> suc B = U.suc B))
21	16, 20	mtbird 717	. . . . . 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 390	. . . 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 2964	. . . . 5 |- (A e. On -> Ord A)
26		orduniorsuc 3093	. . . . . 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 391	. . 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 1483	1 |- ((A e. On /\ A = suc B) -> suc U.A = A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  U.cuni 2507  Ord word 2953  Oncon0 2954  suc csuc 2956

This theorem is referenced by:  rankxplim3 4724  rankxpsuc 4725

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-suc 2960

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