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Theorem moop2 4143
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 2136 . . . 4  |-  ( ( A  =  <. B ,  x >.  /\  A  =  <. [_ y  /  x ]_ B ,  y >.
)  ->  <. B ,  x >.  =  <. [_ y  /  x ]_ B , 
y >. )
2 moop2.1 . . . . . 6  |-  B  e. 
_V
3 vex 2663 . . . . . 6  |-  x  e. 
_V
42, 3opth 4129 . . . . 5  |-  ( <. B ,  x >.  = 
<. [_ y  /  x ]_ B ,  y >.  <->  ( B  =  [_ y  /  x ]_ B  /\  x  =  y )
)
54simprbi 273 . . . 4  |-  ( <. B ,  x >.  = 
<. [_ y  /  x ]_ B ,  y >.  ->  x  =  y )
61, 5syl 14 . . 3  |-  ( ( A  =  <. B ,  x >.  /\  A  =  <. [_ y  /  x ]_ B ,  y >.
)  ->  x  =  y )
76gen2 1411 . 2  |-  A. x A. y ( ( A  =  <. B ,  x >.  /\  A  =  <. [_ y  /  x ]_ B ,  y >. )  ->  x  =  y )
8 nfcsb1v 3005 . . . . 5  |-  F/_ x [_ y  /  x ]_ B
9 nfcv 2258 . . . . 5  |-  F/_ x
y
108, 9nfop 3691 . . . 4  |-  F/_ x <. [_ y  /  x ]_ B ,  y >.
1110nfeq2 2270 . . 3  |-  F/ x  A  =  <. [_ y  /  x ]_ B , 
y >.
12 csbeq1a 2983 . . . . 5  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
13 id 19 . . . . 5  |-  ( x  =  y  ->  x  =  y )
1412, 13opeq12d 3683 . . . 4  |-  ( x  =  y  ->  <. B ,  x >.  =  <. [_ y  /  x ]_ B , 
y >. )
1514eqeq2d 2129 . . 3  |-  ( x  =  y  ->  ( A  =  <. B ,  x >. 
<->  A  =  <. [_ y  /  x ]_ B , 
y >. ) )
1611, 15mo4f 2037 . 2  |-  ( E* x  A  =  <. B ,  x >.  <->  A. x A. y ( ( A  =  <. B ,  x >.  /\  A  =  <. [_ y  /  x ]_ B ,  y >. )  ->  x  =  y ) )
177, 16mpbir 145 1  |-  E* x  A  =  <. B ,  x >.
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1314    = wceq 1316    e. wcel 1465   E*wmo 1978   _Vcvv 2660   [_csb 2975   <.cop 3500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506
This theorem is referenced by: (None)
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