Theorem onminsb 6605

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 3798	. . 3 |- ( { x e. On | ph } =/= (/) <-> E. x e. On ph )
2		ssrab2 3578	. . . 4 |- { x e. On | ph } C_ On
3		onint 6601	. . . 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 3035	. . . . 5 |- F/_ x { x e. On | ph }
7	6	nfint 4285	. . . 4 |- F/_ x |^| { x e. On | ph }
8		nfcv 2622	. . . 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 3252	. . 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 1374   F/wnf 1594   e. wcel 1762   =/= wne 2655   E.wrex 2808   {crab 2811   C_ wss 3469   (/)c0 3778   |^|cint 4275   Oncon0 4871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-br 4441  df-opab 4499  df-tr 4534  df-eprel 4784  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875

This theorem is referenced by:  oawordeulem  7193  rankidb  8207  cardmin2  8368  cardaleph  8459  cardmin  8928