Theorem onnminsb 7004

Description: An ordinal number smaller than the minimum of a set of ordinal numbers does not have the property determining that set. ps is the wff resulting from the substitution of A for x in wff ph. (Contributed by NM, 9-Nov-2003.)

Hypothesis

Ref	Expression
onnminsb.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
onnminsb	|- ( A e. On -> ( A e. |^| { x e. On | ph } -> -. ps ) )

Distinct variable groups:   x, A   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem onnminsb

Step	Hyp	Ref	Expression
1		onnminsb.1	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
2	1	elrab 3363	. . . 4 |- ( A e. { x e. On | ph } <-> ( A e. On /\ ps ) )
3		ssrab2 3687	. . . . 5 |- { x e. On | ph } C_ On
4		onnmin 7003	. . . . 5 |- ( ( { x e. On | ph } C_ On /\ A e. { x e. On | ph } ) -> -. A e. |^| { x e. On | ph } )
5	3, 4	mpan 706	. . . 4 |- ( A e. { x e. On | ph } -> -. A e. |^| { x e. On | ph } )
6	2, 5	sylbir 225	. . 3 |- ( ( A e. On /\ ps ) -> -. A e. |^| { x e. On | ph } )
7	6	ex 450	. 2 |- ( A e. On -> ( ps -> -. A e. |^| { x e. On | ph } ) )
8	7	con2d 129	1 |- ( A e. On -> ( A e. |^| { x e. On | ph } -> -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   |^|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:  onminex  7007  oawordeulem  7634  oeeulem  7681  nnawordex  7717  tcrank  8747  alephnbtwn  8894  cardaleph  8912  cardmin  9386  sltval2  31809  nosepeq  31835  nosupbnd2lem1  31861