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Theorem moop2 4261
Description: "At most one" property of an ordered pair. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypothesis
Ref Expression
moop2.1  |-  B  e. 
_V
Assertion
Ref Expression
moop2  |-  E* x  A  =  <. B ,  x >.
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem moop2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqtr2 2301 . . . 4  |-  ( ( A  =  <. B ,  x >.  /\  A  =  <. [_ y  /  x ]_ B ,  y >.
)  ->  <. B ,  x >.  =  <. [_ y  /  x ]_ B , 
y >. )
2 moop2.1 . . . . . 6  |-  B  e. 
_V
3 vex 2791 . . . . . 6  |-  x  e. 
_V
42, 3opth 4245 . . . . 5  |-  ( <. B ,  x >.  = 
<. [_ y  /  x ]_ B ,  y >.  <->  ( B  =  [_ y  /  x ]_ B  /\  x  =  y )
)
54simprbi 450 . . . 4  |-  ( <. B ,  x >.  = 
<. [_ y  /  x ]_ B ,  y >.  ->  x  =  y )
61, 5syl 15 . . 3  |-  ( ( A  =  <. B ,  x >.  /\  A  =  <. [_ y  /  x ]_ B ,  y >.
)  ->  x  =  y )
76gen2 1534 . 2  |-  A. x A. y ( ( A  =  <. B ,  x >.  /\  A  =  <. [_ y  /  x ]_ B ,  y >. )  ->  x  =  y )
8 nfcsb1v 3113 . . . . 5  |-  F/_ x [_ y  /  x ]_ B
9 nfcv 2419 . . . . 5  |-  F/_ x
y
108, 9nfop 3812 . . . 4  |-  F/_ x <. [_ y  /  x ]_ B ,  y >.
1110nfeq2 2430 . . 3  |-  F/ x  A  =  <. [_ y  /  x ]_ B , 
y >.
12 csbeq1a 3089 . . . . 5  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
13 id 19 . . . . 5  |-  ( x  =  y  ->  x  =  y )
1412, 13opeq12d 3804 . . . 4  |-  ( x  =  y  ->  <. B ,  x >.  =  <. [_ y  /  x ]_ B , 
y >. )
1514eqeq2d 2294 . . 3  |-  ( x  =  y  ->  ( A  =  <. B ,  x >. 
<->  A  =  <. [_ y  /  x ]_ B , 
y >. ) )
1611, 15mo4f 2175 . 2  |-  ( E* x  A  =  <. B ,  x >.  <->  A. x A. y ( ( A  =  <. B ,  x >.  /\  A  =  <. [_ y  /  x ]_ B ,  y >. )  ->  x  =  y ) )
177, 16mpbir 200 1  |-  E* x  A  =  <. B ,  x >.
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   E*wmo 2144   _Vcvv 2788   [_csb 3081   <.cop 3643
This theorem is referenced by:  euop2  4266
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-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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649
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