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Theorem moop2 4392
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 2405 . . . 4  |-  ( ( A  =  <. B ,  x >.  /\  A  =  <. [_ y  /  x ]_ B ,  y >.
)  ->  <. B ,  x >.  =  <. [_ y  /  x ]_ B , 
y >. )
2 moop2.1 . . . . . 6  |-  B  e. 
_V
3 vex 2902 . . . . . 6  |-  x  e. 
_V
42, 3opth 4376 . . . . 5  |-  ( <. B ,  x >.  = 
<. [_ y  /  x ]_ B ,  y >.  <->  ( B  =  [_ y  /  x ]_ B  /\  x  =  y )
)
54simprbi 451 . . . 4  |-  ( <. B ,  x >.  = 
<. [_ y  /  x ]_ B ,  y >.  ->  x  =  y )
61, 5syl 16 . . 3  |-  ( ( A  =  <. B ,  x >.  /\  A  =  <. [_ y  /  x ]_ B ,  y >.
)  ->  x  =  y )
76gen2 1553 . 2  |-  A. x A. y ( ( A  =  <. B ,  x >.  /\  A  =  <. [_ y  /  x ]_ B ,  y >. )  ->  x  =  y )
8 nfcsb1v 3226 . . . . 5  |-  F/_ x [_ y  /  x ]_ B
9 nfcv 2523 . . . . 5  |-  F/_ x
y
108, 9nfop 3942 . . . 4  |-  F/_ x <. [_ y  /  x ]_ B ,  y >.
1110nfeq2 2534 . . 3  |-  F/ x  A  =  <. [_ y  /  x ]_ B , 
y >.
12 csbeq1a 3202 . . . . 5  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
13 id 20 . . . . 5  |-  ( x  =  y  ->  x  =  y )
1412, 13opeq12d 3934 . . . 4  |-  ( x  =  y  ->  <. B ,  x >.  =  <. [_ y  /  x ]_ B , 
y >. )
1514eqeq2d 2398 . . 3  |-  ( x  =  y  ->  ( A  =  <. B ,  x >. 
<->  A  =  <. [_ y  /  x ]_ B , 
y >. ) )
1611, 15mo4f 2270 . 2  |-  ( E* x  A  =  <. B ,  x >.  <->  A. x A. y ( ( A  =  <. B ,  x >.  /\  A  =  <. [_ y  /  x ]_ B ,  y >. )  ->  x  =  y ) )
177, 16mpbir 201 1  |-  E* x  A  =  <. B ,  x >.
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1717   E*wmo 2239   _Vcvv 2899   [_csb 3194   <.cop 3760
This theorem is referenced by:  euop2  4397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766
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