ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  moop2 Unicode version

Theorem moop2 4211
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 2176 . . . 4  |-  ( ( A  =  <. B ,  x >.  /\  A  =  <. [_ y  /  x ]_ B ,  y >.
)  ->  <. B ,  x >.  =  <. [_ y  /  x ]_ B , 
y >. )
2 moop2.1 . . . . . 6  |-  B  e. 
_V
3 vex 2715 . . . . . 6  |-  x  e. 
_V
42, 3opth 4197 . . . . 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 1430 . 2  |-  A. x A. y ( ( A  =  <. B ,  x >.  /\  A  =  <. [_ y  /  x ]_ B ,  y >. )  ->  x  =  y )
8 nfcsb1v 3064 . . . . 5  |-  F/_ x [_ y  /  x ]_ B
9 nfcv 2299 . . . . 5  |-  F/_ x
y
108, 9nfop 3757 . . . 4  |-  F/_ x <. [_ y  /  x ]_ B ,  y >.
1110nfeq2 2311 . . 3  |-  F/ x  A  =  <. [_ y  /  x ]_ B , 
y >.
12 csbeq1a 3040 . . . . 5  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
13 id 19 . . . . 5  |-  ( x  =  y  ->  x  =  y )
1412, 13opeq12d 3749 . . . 4  |-  ( x  =  y  ->  <. B ,  x >.  =  <. [_ y  /  x ]_ B , 
y >. )
1514eqeq2d 2169 . . 3  |-  ( x  =  y  ->  ( A  =  <. B ,  x >. 
<->  A  =  <. [_ y  /  x ]_ B , 
y >. ) )
1611, 15mo4f 2066 . 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 1333    = wceq 1335   E*wmo 2007    e. wcel 2128   _Vcvv 2712   [_csb 3031   <.cop 3563
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator