Theorem onminsb 6633

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 3814	. . 3 |- ( { x e. On | ph } =/= (/) <-> E. x e. On ph )
2		ssrab2 3581	. . . 4 |- { x e. On | ph } C_ On
3		onint 6629	. . . 4 |- ( ( { x e. On | ph } C_ On /\ { x e. On | ph } =/= (/) ) -> |^| { x e. On | ph } e. { x e. On | ph } )
4	2, 3	mpan 670	. . 3 |- ( { x e. On | ph } =/= (/) -> |^| { x e. On | ph } e. { x e. On | ph } )
5	1, 4	sylbir 213	. 2 |- ( E. x e. On ph -> |^| { x e. On | ph } e. { x e. On | ph } )
6		nfrab1 3038	. . . . 5 |- F/_ x { x e. On | ph }
7	6	nfint 4298	. . . 4 |- F/_ x |^| { x e. On | ph }
8		nfcv 2619	. . . 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 3255	. . 3 |- ( |^| { x e. On | ph } e. { x e. On | ph } <-> ( |^| { x e. On | ph } e. On /\ ps ) )
12	11	simprbi 464	. 2 |- ( |^| { x e. On | ph } e. { x e. On | ph } -> ps )
13	5, 12	syl 16	1 |- ( E. x e. On ph -> ps )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1395   F/wnf 1617   e. wcel 1819   =/= wne 2652   E.wrex 2808   {crab 2811   C_ wss 3471   (/)c0 3793   |^|cint 4288   Oncon0 4887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891

This theorem is referenced by:  oawordeulem  7221  rankidb  8235  cardmin2  8396  cardaleph  8487  cardmin  8956