Theorem ordsucuniel 7024

Description: Given an element A of the union of an ordinal B, suc A is an element of B itself. (Contributed by Scott Fenton, 28-Mar-2012.) (Proof shortened by Mario Carneiro, 29-May-2015.)

Assertion

Ref	Expression
ordsucuniel	|- ( Ord B -> ( A e. U. B <-> suc A e. B ) )

Proof of Theorem ordsucuniel

Step	Hyp	Ref	Expression
1		orduni 6994	. . 3 |- ( Ord B -> Ord U. B )
2		ordelord 5745	. . . 4 |- ( ( Ord U. B /\ A e. U. B ) -> Ord A )
3	2	ex 450	. . 3 |- ( Ord U. B -> ( A e. U. B -> Ord A ) )
4	1, 3	syl 17	. 2 |- ( Ord B -> ( A e. U. B -> Ord A ) )
5		ordelord 5745	. . . 4 |- ( ( Ord B /\ suc A e. B ) -> Ord suc A )
6		ordsuc 7014	. . . 4 |- ( Ord A <-> Ord suc A )
7	5, 6	sylibr 224	. . 3 |- ( ( Ord B /\ suc A e. B ) -> Ord A )
8	7	ex 450	. 2 |- ( Ord B -> ( suc A e. B -> Ord A ) )
9		ordsson 6989	. . . . . 6 |- ( Ord B -> B C_ On )
10		ordunisssuc 5830	. . . . . 6 |- ( ( B C_ On /\ Ord A ) -> ( U. B C_ A <-> B C_ suc A ) )
11	9, 10	sylan 488	. . . . 5 |- ( ( Ord B /\ Ord A ) -> ( U. B C_ A <-> B C_ suc A ) )
12		ordtri1 5756	. . . . . 6 |- ( ( Ord U. B /\ Ord A ) -> ( U. B C_ A <-> -. A e. U. B ) )
13	1, 12	sylan 488	. . . . 5 |- ( ( Ord B /\ Ord A ) -> ( U. B C_ A <-> -. A e. U. B ) )
14		ordtri1 5756	. . . . . 6 |- ( ( Ord B /\ Ord suc A ) -> ( B C_ suc A <-> -. suc A e. B ) )
15	6, 14	sylan2b 492	. . . . 5 |- ( ( Ord B /\ Ord A ) -> ( B C_ suc A <-> -. suc A e. B ) )
16	11, 13, 15	3bitr3d 298	. . . 4 |- ( ( Ord B /\ Ord A ) -> ( -. A e. U. B <-> -. suc A e. B ) )
17	16	con4bid 307	. . 3 |- ( ( Ord B /\ Ord A ) -> ( A e. U. B <-> suc A e. B ) )
18	17	ex 450	. 2 |- ( Ord B -> ( Ord A -> ( A e. U. B <-> suc A e. B ) ) )
19	4, 8, 18	pm5.21ndd 369	1 |- ( Ord B -> ( A e. U. B <-> suc A e. B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   C_ wss 3574   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:  dfac12lem1  8965  dfac12lem2  8966