Theorem ordtoplem 32434

Description: Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.)

Hypothesis

Ref	Expression
ordtoplem.1	|- ( U. A e. On -> suc U. A e. S )

Assertion

Ref	Expression
ordtoplem	|- ( Ord A -> ( A =/= U. A -> A e. S ) )

Proof of Theorem ordtoplem

Step	Hyp	Ref	Expression
1		df-ne 2795	. 2 |- ( A =/= U. A <-> -. A = U. A )
2		ordeleqon 6988	. . . . . 6 |- ( Ord A <-> ( A e. On \/ A = On ) )
3		unon 7031	. . . . . . . . 9 |- U. On = On
4	3	eqcomi 2631	. . . . . . . 8 |- On = U. On
5		id 22	. . . . . . . 8 |- ( A = On -> A = On )
6		unieq 4444	. . . . . . . 8 |- ( A = On -> U. A = U. On )
7	4, 5, 6	3eqtr4a 2682	. . . . . . 7 |- ( A = On -> A = U. A )
8	7	orim2i 540	. . . . . 6 |- ( ( A e. On \/ A = On ) -> ( A e. On \/ A = U. A ) )
9	2, 8	sylbi 207	. . . . 5 |- ( Ord A -> ( A e. On \/ A = U. A ) )
10	9	orcomd 403	. . . 4 |- ( Ord A -> ( A = U. A \/ A e. On ) )
11	10	ord 392	. . 3 |- ( Ord A -> ( -. A = U. A -> A e. On ) )
12		orduniorsuc 7030	. . . 4 |- ( Ord A -> ( A = U. A \/ A = suc U. A ) )
13	12	ord 392	. . 3 |- ( Ord A -> ( -. A = U. A -> A = suc U. A ) )
14		onuni 6993	. . . 4 |- ( A e. On -> U. A e. On )
15		ordtoplem.1	. . . 4 |- ( U. A e. On -> suc U. A e. S )
16		eleq1a 2696	. . . 4 |- ( suc U. A e. S -> ( A = suc U. A -> A e. S ) )
17	14, 15, 16	3syl 18	. . 3 |- ( A e. On -> ( A = suc U. A -> A e. S ) )
18	11, 13, 17	syl6c 70	. 2 |- ( Ord A -> ( -. A = U. A -> A e. S ) )
19	1, 18	syl5bi 232	1 |- ( Ord A -> ( A =/= U. A -> A e. S ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   = wceq 1483   e. wcel 1990   =/= wne 2794   U.cuni 4436   Ord word 5722   Oncon0 5723   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-suc 5729

This theorem is referenced by:  ordtop  32435  ordtopconn  32438  ordtopt0  32441