MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  onminex Structured version   Unicode version

Theorem onminex 6637
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 3590 . . . 4  |-  { x  e.  On  |  ph }  C_  On
2 rabn0 3810 . . . . 5  |-  ( { x  e.  On  |  ph }  =/=  (/)  <->  E. x  e.  On  ph )
32biimpri 206 . . . 4  |-  ( E. x  e.  On  ph  ->  { x  e.  On  |  ph }  =/=  (/) )
4 oninton 6630 . . . 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 6628 . . 3  |-  ( E. x  e.  On  ph  ->  [. |^| { x  e.  On  |  ph }  /  x ]. ph )
7 onss 6621 . . . . . . 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 3508 . . . . 5  |-  ( E. x  e.  On  ph  ->  ( y  e.  |^| { x  e.  On  |  ph }  ->  y  e.  On ) )
10 onminex.1 . . . . . 6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
1110onnminsb 6634 . . . . 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 2879 . . 3  |-  ( E. x  e.  On  ph  ->  A. y  e.  |^| { x  e.  On  |  ph }  -.  ps )
14 dfsbcq2 3339 . . . . 5  |-  ( z  =  |^| { x  e.  On  |  ph }  ->  ( [ z  /  x ] ph  <->  [. |^| { x  e.  On  |  ph }  /  x ]. ph )
)
15 raleq 3063 . . . . 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 3219 . . 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 1226 . 2  |-  ( E. x  e.  On  ph  ->  E. z  e.  On  ( [ z  /  x ] ph  /\  A. y  e.  z  -.  ps )
)
19 nfv 1683 . . 3  |-  F/ z ( ph  /\  A. y  e.  x  -.  ps )
20 nfs1v 2164 . . . 4  |-  F/ x [ z  /  x ] ph
21 nfv 1683 . . . 4  |-  F/ x A. y  e.  z  -.  ps
2220, 21nfan 1875 . . 3  |-  F/ x
( [ z  /  x ] ph  /\  A. y  e.  z  -.  ps )
23 sbequ12 1961 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
24 raleq 3063 . . . 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 3090 . 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 1379   [wsb 1711    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818   {crab 2821   [.wsbc 3336    C_ wss 3481   (/)c0 3790   |^|cint 4288   Oncon0 4884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-br 4454  df-opab 4512  df-tr 4547  df-eprel 4797  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888
This theorem is referenced by:  tz7.49  7122  omeulem1  7243  zorn2lem7  8894
  Copyright terms: Public domain W3C validator