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Theorem onminex 6518
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 3535 . . . 4  |-  { x  e.  On  |  ph }  C_  On
2 rabn0 3755 . . . . 5  |-  ( { x  e.  On  |  ph }  =/=  (/)  <->  E. x  e.  On  ph )
32biimpri 206 . . . 4  |-  ( E. x  e.  On  ph  ->  { x  e.  On  |  ph }  =/=  (/) )
4 oninton 6511 . . . 4  |-  ( ( { x  e.  On  |  ph }  C_  On  /\ 
{ x  e.  On  |  ph }  =/=  (/) )  ->  |^| { x  e.  On  |  ph }  e.  On )
51, 3, 4sylancr 663 . . 3  |-  ( E. x  e.  On  ph  ->  |^| { x  e.  On  |  ph }  e.  On )
6 onminesb 6509 . . 3  |-  ( E. x  e.  On  ph  ->  [. |^| { x  e.  On  |  ph }  /  x ]. ph )
7 onss 6502 . . . . . . 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 3453 . . . . 5  |-  ( E. x  e.  On  ph  ->  ( y  e.  |^| { x  e.  On  |  ph }  ->  y  e.  On ) )
10 onminex.1 . . . . . 6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
1110onnminsb 6515 . . . . 5  |-  ( y  e.  On  ->  (
y  e.  |^| { x  e.  On  |  ph }  ->  -.  ps ) )
129, 11syli 37 . . . 4  |-  ( E. x  e.  On  ph  ->  ( y  e.  |^| { x  e.  On  |  ph }  ->  -.  ps )
)
1312ralrimiv 2820 . . 3  |-  ( E. x  e.  On  ph  ->  A. y  e.  |^| { x  e.  On  |  ph }  -.  ps )
14 dfsbcq2 3287 . . . . 5  |-  ( z  =  |^| { x  e.  On  |  ph }  ->  ( [ z  /  x ] ph  <->  [. |^| { x  e.  On  |  ph }  /  x ]. ph )
)
15 raleq 3013 . . . . 5  |-  ( z  =  |^| { x  e.  On  |  ph }  ->  ( A. y  e.  z  -.  ps  <->  A. y  e.  |^| { x  e.  On  |  ph }  -.  ps ) )
1614, 15anbi12d 710 . . . 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 3169 . . 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 1217 . 2  |-  ( E. x  e.  On  ph  ->  E. z  e.  On  ( [ z  /  x ] ph  /\  A. y  e.  z  -.  ps )
)
19 nfv 1674 . . 3  |-  F/ z ( ph  /\  A. y  e.  x  -.  ps )
20 nfs1v 2149 . . . 4  |-  F/ x [ z  /  x ] ph
21 nfv 1674 . . . 4  |-  F/ x A. y  e.  z  -.  ps
2220, 21nfan 1863 . . 3  |-  F/ x
( [ z  /  x ] ph  /\  A. y  e.  z  -.  ps )
23 sbequ12 1945 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
24 raleq 3013 . . . 4  |-  ( x  =  z  ->  ( A. y  e.  x  -.  ps  <->  A. y  e.  z  -.  ps ) )
2523, 24anbi12d 710 . . 3  |-  ( x  =  z  ->  (
( ph  /\  A. y  e.  x  -.  ps )  <->  ( [ z  /  x ] ph  /\  A. y  e.  z  -.  ps )
) )
2619, 22, 25cbvrex 3040 . 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 212 1  |-  ( E. x  e.  On  ph  ->  E. x  e.  On  ( ph  /\  A. y  e.  x  -.  ps )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370   [wsb 1702    e. wcel 1758    =/= wne 2644   A.wral 2795   E.wrex 2796   {crab 2799   [.wsbc 3284    C_ wss 3426   (/)c0 3735   |^|cint 4226   Oncon0 4817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-br 4391  df-opab 4449  df-tr 4484  df-eprel 4730  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821
This theorem is referenced by:  tz7.49  7000  omeulem1  7121  zorn2lem7  8772
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