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Theorem onminex 4598
Description: If a wff is true for an ordinal number, there is a 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 3258 . . . 4  |-  { x  e.  On  |  ph }  C_  On
2 rabn0 3474 . . . . 5  |-  ( { x  e.  On  |  ph }  =/=  (/)  <->  E. x  e.  On  ph )
32biimpri 197 . . . 4  |-  ( E. x  e.  On  ph  ->  { x  e.  On  |  ph }  =/=  (/) )
4 oninton 4591 . . . 4  |-  ( ( { x  e.  On  |  ph }  C_  On  /\ 
{ x  e.  On  |  ph }  =/=  (/) )  ->  |^| { x  e.  On  |  ph }  e.  On )
51, 3, 4sylancr 644 . . 3  |-  ( E. x  e.  On  ph  ->  |^| { x  e.  On  |  ph }  e.  On )
6 onminesb 4589 . . 3  |-  ( E. x  e.  On  ph  ->  [. |^| { x  e.  On  |  ph }  /  x ]. ph )
7 onss 4582 . . . . . . 7  |-  ( |^| { x  e.  On  |  ph }  e.  On  ->  |^|
{ x  e.  On  |  ph }  C_  On )
85, 7syl 15 . . . . . 6  |-  ( E. x  e.  On  ph  ->  |^| { x  e.  On  |  ph }  C_  On )
98sseld 3179 . . . . 5  |-  ( E. x  e.  On  ph  ->  ( y  e.  |^| { x  e.  On  |  ph }  ->  y  e.  On ) )
10 onminex.1 . . . . . 6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
1110onnminsb 4595 . . . . 5  |-  ( y  e.  On  ->  (
y  e.  |^| { x  e.  On  |  ph }  ->  -.  ps ) )
129, 11syli 33 . . . 4  |-  ( E. x  e.  On  ph  ->  ( y  e.  |^| { x  e.  On  |  ph }  ->  -.  ps )
)
1312ralrimiv 2625 . . 3  |-  ( E. x  e.  On  ph  ->  A. y  e.  |^| { x  e.  On  |  ph }  -.  ps )
14 dfsbcq2 2994 . . . . 5  |-  ( z  =  |^| { x  e.  On  |  ph }  ->  ( [ z  /  x ] ph  <->  [. |^| { x  e.  On  |  ph }  /  x ]. ph )
)
15 raleq 2736 . . . . 5  |-  ( z  =  |^| { x  e.  On  |  ph }  ->  ( A. y  e.  z  -.  ps  <->  A. y  e.  |^| { x  e.  On  |  ph }  -.  ps ) )
1614, 15anbi12d 691 . . . 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 2884 . . 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 1180 . 2  |-  ( E. x  e.  On  ph  ->  E. z  e.  On  ( [ z  /  x ] ph  /\  A. y  e.  z  -.  ps )
)
19 nfv 1605 . . 3  |-  F/ z ( ph  /\  A. y  e.  x  -.  ps )
20 nfs1v 2045 . . . 4  |-  F/ x [ z  /  x ] ph
21 nfv 1605 . . . 4  |-  F/ x A. y  e.  z  -.  ps
2220, 21nfan 1771 . . 3  |-  F/ x
( [ z  /  x ] ph  /\  A. y  e.  z  -.  ps )
23 sbequ12 1860 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
24 raleq 2736 . . . 4  |-  ( x  =  z  ->  ( A. y  e.  x  -.  ps  <->  A. y  e.  z  -.  ps ) )
2523, 24anbi12d 691 . . 3  |-  ( x  =  z  ->  (
( ph  /\  A. y  e.  x  -.  ps )  <->  ( [ z  /  x ] ph  /\  A. y  e.  z  -.  ps )
) )
2619, 22, 25cbvrex 2761 . 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 203 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 176    /\ wa 358    = wceq 1623   [wsb 1629    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   [.wsbc 2991    C_ wss 3152   (/)c0 3455   |^|cint 3862   Oncon0 4392
This theorem is referenced by:  tz7.49  6457  omeulem1  6580  zorn2lem7  8129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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