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Theorem onminex 6592
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 3489 . . . 4  |-  { x  e.  On  |  ph }  C_  On
2 rabn0 3725 . . . . 5  |-  ( { x  e.  On  |  ph }  =/=  (/)  <->  E. x  e.  On  ph )
32biimpri 209 . . . 4  |-  ( E. x  e.  On  ph  ->  { x  e.  On  |  ph }  =/=  (/) )
4 oninton 6585 . . . 4  |-  ( ( { x  e.  On  |  ph }  C_  On  /\ 
{ x  e.  On  |  ph }  =/=  (/) )  ->  |^| { x  e.  On  |  ph }  e.  On )
51, 3, 4sylancr 667 . . 3  |-  ( E. x  e.  On  ph  ->  |^| { x  e.  On  |  ph }  e.  On )
6 onminesb 6583 . . 3  |-  ( E. x  e.  On  ph  ->  [. |^| { x  e.  On  |  ph }  /  x ]. ph )
7 onss 6575 . . . . . . 7  |-  ( |^| { x  e.  On  |  ph }  e.  On  ->  |^|
{ x  e.  On  |  ph }  C_  On )
85, 7syl 17 . . . . . 6  |-  ( E. x  e.  On  ph  ->  |^| { x  e.  On  |  ph }  C_  On )
98sseld 3406 . . . . 5  |-  ( E. x  e.  On  ph  ->  ( y  e.  |^| { x  e.  On  |  ph }  ->  y  e.  On ) )
10 onminex.1 . . . . . 6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
1110onnminsb 6589 . . . . 5  |-  ( y  e.  On  ->  (
y  e.  |^| { x  e.  On  |  ph }  ->  -.  ps ) )
129, 11syli 38 . . . 4  |-  ( E. x  e.  On  ph  ->  ( y  e.  |^| { x  e.  On  |  ph }  ->  -.  ps )
)
1312ralrimiv 2777 . . 3  |-  ( E. x  e.  On  ph  ->  A. y  e.  |^| { x  e.  On  |  ph }  -.  ps )
14 dfsbcq2 3245 . . . . 5  |-  ( z  =  |^| { x  e.  On  |  ph }  ->  ( [ z  /  x ] ph  <->  [. |^| { x  e.  On  |  ph }  /  x ]. ph )
)
15 raleq 2964 . . . . 5  |-  ( z  =  |^| { x  e.  On  |  ph }  ->  ( A. y  e.  z  -.  ps  <->  A. y  e.  |^| { x  e.  On  |  ph }  -.  ps ) )
1614, 15anbi12d 715 . . . 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 3125 . . 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 1262 . 2  |-  ( E. x  e.  On  ph  ->  E. z  e.  On  ( [ z  /  x ] ph  /\  A. y  e.  z  -.  ps )
)
19 nfv 1755 . . 3  |-  F/ z ( ph  /\  A. y  e.  x  -.  ps )
20 nfs1v 2243 . . . 4  |-  F/ x [ z  /  x ] ph
21 nfv 1755 . . . 4  |-  F/ x A. y  e.  z  -.  ps
2220, 21nfan 1988 . . 3  |-  F/ x
( [ z  /  x ] ph  /\  A. y  e.  z  -.  ps )
23 sbequ12 2057 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
24 raleq 2964 . . . 4  |-  ( x  =  z  ->  ( A. y  e.  x  -.  ps  <->  A. y  e.  z  -.  ps ) )
2523, 24anbi12d 715 . . 3  |-  ( x  =  z  ->  (
( ph  /\  A. y  e.  x  -.  ps )  <->  ( [ z  /  x ] ph  /\  A. y  e.  z  -.  ps )
) )
2619, 22, 25cbvrex 2993 . 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 215 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 187    /\ wa 370    = wceq 1437   [wsb 1790    e. wcel 1872    =/= wne 2599   A.wral 2714   E.wrex 2715   {crab 2718   [.wsbc 3242    C_ wss 3379   (/)c0 3704   |^|cint 4198   Oncon0 5385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-br 4367  df-opab 4426  df-tr 4462  df-eprel 4707  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-ord 5388  df-on 5389
This theorem is referenced by:  tz7.49  7117  omeulem1  7238  zorn2lem7  8883
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