Theorem onminsb 6999

Description: If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses implicit substitution. Theorem Schema 62 of [Suppes] p. 228. (Contributed by NM, 3-Oct-2003.)

Hypotheses

Ref	Expression
onminsb.1	|- F/ x ps
onminsb.2	|- ( x = |^| { x e. On | ph } -> ( ph <-> ps ) )

Assertion

Ref	Expression
onminsb	|- ( E. x e. On ph -> ps )

Proof of Theorem onminsb

Step	Hyp	Ref	Expression
1		rabn0 3958	. . 3 |- ( { x e. On | ph } =/= (/) <-> E. x e. On ph )
2		ssrab2 3687	. . . 4 |- { x e. On | ph } C_ On
3		onint 6995	. . . 4 |- ( ( { x e. On | ph } C_ On /\ { x e. On | ph } =/= (/) ) -> |^| { x e. On | ph } e. { x e. On | ph } )
4	2, 3	mpan 706	. . 3 |- ( { x e. On | ph } =/= (/) -> |^| { x e. On | ph } e. { x e. On | ph } )
5	1, 4	sylbir 225	. 2 |- ( E. x e. On ph -> |^| { x e. On | ph } e. { x e. On | ph } )
6		nfrab1 3122	. . . . 5 |- F/_ x { x e. On | ph }
7	6	nfint 4486	. . . 4 |- F/_ x |^| { x e. On | ph }
8		nfcv 2764	. . . 4 |- F/_ x On
9		onminsb.1	. . . 4 |- F/ x ps
10		onminsb.2	. . . 4 |- ( x = |^| { x e. On | ph } -> ( ph <-> ps ) )
11	7, 8, 9, 10	elrabf 3360	. . 3 |- ( |^| { x e. On | ph } e. { x e. On | ph } <-> ( |^| { x e. On | ph } e. On /\ ps ) )
12	11	simprbi 480	. 2 |- ( |^| { x e. On | ph } e. { x e. On | ph } -> ps )
13	5, 12	syl 17	1 |- ( E. x e. On ph -> ps )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   F/wnf 1708   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   |^|cint 4475   Oncon0 5723

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-int 4476  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

This theorem is referenced by:  oawordeulem  7634  rankidb  8663  cardmin2  8824  cardaleph  8912  cardmin  9386