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Theorem moop2 4454
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 2456 . . . 4  |-  ( ( A  =  <. B ,  x >.  /\  A  =  <. [_ y  /  x ]_ B ,  y >.
)  ->  <. B ,  x >.  =  <. [_ y  /  x ]_ B , 
y >. )
2 moop2.1 . . . . . 6  |-  B  e. 
_V
3 vex 2961 . . . . . 6  |-  x  e. 
_V
42, 3opth 4438 . . . . 5  |-  ( <. B ,  x >.  = 
<. [_ y  /  x ]_ B ,  y >.  <->  ( B  =  [_ y  /  x ]_ B  /\  x  =  y )
)
54simprbi 452 . . . 4  |-  ( <. B ,  x >.  = 
<. [_ y  /  x ]_ B ,  y >.  ->  x  =  y )
61, 5syl 16 . . 3  |-  ( ( A  =  <. B ,  x >.  /\  A  =  <. [_ y  /  x ]_ B ,  y >.
)  ->  x  =  y )
76gen2 1557 . 2  |-  A. x A. y ( ( A  =  <. B ,  x >.  /\  A  =  <. [_ y  /  x ]_ B ,  y >. )  ->  x  =  y )
8 nfcsb1v 3285 . . . . 5  |-  F/_ x [_ y  /  x ]_ B
9 nfcv 2574 . . . . 5  |-  F/_ x
y
108, 9nfop 4002 . . . 4  |-  F/_ x <. [_ y  /  x ]_ B ,  y >.
1110nfeq2 2585 . . 3  |-  F/ x  A  =  <. [_ y  /  x ]_ B , 
y >.
12 csbeq1a 3261 . . . . 5  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
13 id 21 . . . . 5  |-  ( x  =  y  ->  x  =  y )
1412, 13opeq12d 3994 . . . 4  |-  ( x  =  y  ->  <. B ,  x >.  =  <. [_ y  /  x ]_ B , 
y >. )
1514eqeq2d 2449 . . 3  |-  ( x  =  y  ->  ( A  =  <. B ,  x >. 
<->  A  =  <. [_ y  /  x ]_ B , 
y >. ) )
1611, 15mo4f 2315 . 2  |-  ( E* x  A  =  <. B ,  x >.  <->  A. x A. y ( ( A  =  <. B ,  x >.  /\  A  =  <. [_ y  /  x ]_ B ,  y >. )  ->  x  =  y ) )
177, 16mpbir 202 1  |-  E* x  A  =  <. B ,  x >.
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726   E*wmo 2284   _Vcvv 2958   [_csb 3253   <.cop 3819
This theorem is referenced by:  euop2  4459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825
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