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Theorem onminex 4788
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.
StepHypRef Expression
1 ssrab2 3429 . . . 4  |-  { x  e.  On  |  ph }  C_  On
2 rabn0 3648 . . . . 5  |-  ( { x  e.  On  |  ph }  =/=  (/)  <->  E. x  e.  On  ph )
32biimpri 199 . . . 4  |-  ( E. x  e.  On  ph  ->  { x  e.  On  |  ph }  =/=  (/) )
4 oninton 4781 . . . 4  |-  ( ( { x  e.  On  |  ph }  C_  On  /\ 
{ x  e.  On  |  ph }  =/=  (/) )  ->  |^| { x  e.  On  |  ph }  e.  On )
51, 3, 4sylancr 646 . . 3  |-  ( E. x  e.  On  ph  ->  |^| { x  e.  On  |  ph }  e.  On )
6 onminesb 4779 . . 3  |-  ( E. x  e.  On  ph  ->  [. |^| { x  e.  On  |  ph }  /  x ]. ph )
7 onss 4772 . . . . . . 7  |-  ( |^| { x  e.  On  |  ph }  e.  On  ->  |^|
{ x  e.  On  |  ph }  C_  On )
85, 7syl 16 . . . . . 6  |-  ( E. x  e.  On  ph  ->  |^| { x  e.  On  |  ph }  C_  On )
98sseld 3348 . . . . 5  |-  ( E. x  e.  On  ph  ->  ( y  e.  |^| { x  e.  On  |  ph }  ->  y  e.  On ) )
10 onminex.1 . . . . . 6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
1110onnminsb 4785 . . . . 5  |-  ( y  e.  On  ->  (
y  e.  |^| { x  e.  On  |  ph }  ->  -.  ps ) )
129, 11syli 36 . . . 4  |-  ( E. x  e.  On  ph  ->  ( y  e.  |^| { x  e.  On  |  ph }  ->  -.  ps )
)
1312ralrimiv 2789 . . 3  |-  ( E. x  e.  On  ph  ->  A. y  e.  |^| { x  e.  On  |  ph }  -.  ps )
14 dfsbcq2 3165 . . . . 5  |-  ( z  =  |^| { x  e.  On  |  ph }  ->  ( [ z  /  x ] ph  <->  [. |^| { x  e.  On  |  ph }  /  x ]. ph )
)
15 raleq 2905 . . . . 5  |-  ( z  =  |^| { x  e.  On  |  ph }  ->  ( A. y  e.  z  -.  ps  <->  A. y  e.  |^| { x  e.  On  |  ph }  -.  ps ) )
1614, 15anbi12d 693 . . . 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 ) ) )
1716rspcev 3053 . . 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 )
)
185, 6, 13, 17syl12anc 1183 . 2  |-  ( E. x  e.  On  ph  ->  E. z  e.  On  ( [ z  /  x ] ph  /\  A. y  e.  z  -.  ps )
)
19 nfv 1630 . . 3  |-  F/ z ( ph  /\  A. y  e.  x  -.  ps )
20 nfs1v 2183 . . . 4  |-  F/ x [ z  /  x ] ph
21 nfv 1630 . . . 4  |-  F/ x A. y  e.  z  -.  ps
2220, 21nfan 1847 . . 3  |-  F/ x
( [ z  /  x ] ph  /\  A. y  e.  z  -.  ps )
23 sbequ12 1945 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
24 raleq 2905 . . . 4  |-  ( x  =  z  ->  ( A. y  e.  x  -.  ps  <->  A. y  e.  z  -.  ps ) )
2523, 24anbi12d 693 . . 3  |-  ( x  =  z  ->  (
( ph  /\  A. y  e.  x  -.  ps )  <->  ( [ z  /  x ] ph  /\  A. y  e.  z  -.  ps )
) )
2619, 22, 25cbvrex 2930 . 2  |-  ( E. x  e.  On  ( ph  /\  A. y  e.  x  -.  ps )  <->  E. z  e.  On  ( [ z  /  x ] ph  /\  A. y  e.  z  -.  ps )
)
2718, 26sylibr 205 1  |-  ( E. x  e.  On  ph  ->  E. x  e.  On  ( ph  /\  A. y  e.  x  -.  ps )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653   [wsb 1659    e. wcel 1726    =/= wne 2600   A.wral 2706   E.wrex 2707   {crab 2710   [.wsbc 3162    C_ wss 3321   (/)c0 3629   |^|cint 4051   Oncon0 4582
This theorem is referenced by:  tz7.49  6703  omeulem1  6826  zorn2lem7  8383
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-br 4214  df-opab 4268  df-tr 4304  df-eprel 4495  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586
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