Theorem onminex 7007

Description: If a wff is true for an ordinal number, there is the smallest ordinal number for which it is true. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Mario Carneiro, 20-Nov-2016.)

Hypothesis

Ref	Expression
onminex.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
onminex	|- ( E. x e. On ph -> E. x e. On ( ph /\ A. y e. x -. ps ) )

Distinct variable groups:   x, y   ph, y   ps, x

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem onminex

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3687	. . . 4 |- { x e. On | ph } C_ On
2		rabn0 3958	. . . . 5 |- ( { x e. On | ph } =/= (/) <-> E. x e. On ph )
3	2	biimpri 218	. . . 4 |- ( E. x e. On ph -> { x e. On | ph } =/= (/) )
4		oninton 7000	. . . 4 |- ( ( { x e. On | ph } C_ On /\ { x e. On | ph } =/= (/) ) -> |^| { x e. On | ph } e. On )
5	1, 3, 4	sylancr 695	. . 3 |- ( E. x e. On ph -> |^| { x e. On | ph } e. On )
6		onminesb 6998	. . 3 |- ( E. x e. On ph -> [. |^| { x e. On | ph } / x ]. ph )
7		onss 6990	. . . . . . 7 |- ( |^| { x e. On | ph } e. On -> |^| { x e. On | ph } C_ On )
8	5, 7	syl 17	. . . . . 6 |- ( E. x e. On ph -> |^| { x e. On | ph } C_ On )
9	8	sseld 3602	. . . . 5 |- ( E. x e. On ph -> ( y e. |^| { x e. On | ph } -> y e. On ) )
10		onminex.1	. . . . . 6 |- ( x = y -> ( ph <-> ps ) )
11	10	onnminsb 7004	. . . . 5 |- ( y e. On -> ( y e. |^| { x e. On | ph } -> -. ps ) )
12	9, 11	syli 39	. . . 4 |- ( E. x e. On ph -> ( y e. |^| { x e. On | ph } -> -. ps ) )
13	12	ralrimiv 2965	. . 3 |- ( E. x e. On ph -> A. y e. |^| { x e. On | ph } -. ps )
14		dfsbcq2 3438	. . . . 5 |- ( z = |^| { x e. On | ph } -> ( [ z / x ] ph <-> [. |^| { x e. On | ph } / x ]. ph ) )
15		raleq 3138	. . . . 5 |- ( z = |^| { x e. On | ph } -> ( A. y e. z -. ps <-> A. y e. |^| { x e. On | ph } -. ps ) )
16	14, 15	anbi12d 747	. . . 4 |- ( z = |^| { x e. On | ph } -> ( ( [ z / x ] ph /\ A. y e. z -. ps ) <-> ( [. |^| { x e. On | ph } / x ]. ph /\ A. y e. |^| { x e. On | ph } -. ps ) ) )
17	16	rspcev 3309	. . 3 |- ( ( |^| { x e. On | ph } e. On /\ ( [. |^| { x e. On | ph } / x ]. ph /\ A. y e. |^| { x e. On | ph } -. ps ) ) -> E. z e. On ( [ z / x ] ph /\ A. y e. z -. ps ) )
18	5, 6, 13, 17	syl12anc 1324	. 2 |- ( E. x e. On ph -> E. z e. On ( [ z / x ] ph /\ A. y e. z -. ps ) )
19		nfv 1843	. . 3 |- F/ z ( ph /\ A. y e. x -. ps )
20		nfs1v 2437	. . . 4 |- F/ x [ z / x ] ph
21		nfv 1843	. . . 4 |- F/ x A. y e. z -. ps
22	20, 21	nfan 1828	. . 3 |- F/ x ( [ z / x ] ph /\ A. y e. z -. ps )
23		sbequ12 2111	. . . 4 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
24		raleq 3138	. . . 4 |- ( x = z -> ( A. y e. x -. ps <-> A. y e. z -. ps ) )
25	23, 24	anbi12d 747	. . 3 |- ( x = z -> ( ( ph /\ A. y e. x -. ps ) <-> ( [ z / x ] ph /\ A. y e. z -. ps ) ) )
26	19, 22, 25	cbvrex 3168	. 2 |- ( E. x e. On ( ph /\ A. y e. x -. ps ) <-> E. z e. On ( [ z / x ] ph /\ A. y e. z -. ps ) )
27	18, 26	sylibr 224	1 |- ( E. x e. On ph -> E. x e. On ( ph /\ A. y e. x -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   [wsb 1880   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   [.wsbc 3435   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:  tz7.49  7540  omeulem1  7662  zorn2lem7  9324