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Theorem xpsnen 6386
Description: A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
Hypotheses
Ref Expression
xpsnen.1  |-  A  e. 
_V
xpsnen.2  |-  B  e. 
_V
Assertion
Ref Expression
xpsnen  |-  ( A  X.  { B }
)  ~~  A

Proof of Theorem xpsnen
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpsnen.1 . . 3  |-  A  e. 
_V
2 xpsnen.2 . . . 4  |-  B  e. 
_V
32snex 3977 . . 3  |-  { B }  e.  _V
41, 3xpex 4501 . 2  |-  ( A  X.  { B }
)  e.  _V
5 elxp 4408 . . 3  |-  ( y  e.  ( A  X.  { B } )  <->  E. x E. z ( y  = 
<. x ,  z >.  /\  ( x  e.  A  /\  z  e.  { B } ) ) )
6 inteq 3659 . . . . . . . 8  |-  ( y  =  <. x ,  z
>.  ->  |^| y  =  |^| <.
x ,  z >.
)
76inteqd 3661 . . . . . . 7  |-  ( y  =  <. x ,  z
>.  ->  |^| |^| y  =  |^| |^|
<. x ,  z >.
)
8 vex 2613 . . . . . . . 8  |-  x  e. 
_V
9 vex 2613 . . . . . . . 8  |-  z  e. 
_V
108, 9op1stb 4255 . . . . . . 7  |-  |^| |^| <. x ,  z >.  =  x
117, 10syl6eq 2131 . . . . . 6  |-  ( y  =  <. x ,  z
>.  ->  |^| |^| y  =  x )
1211, 8syl6eqel 2173 . . . . 5  |-  ( y  =  <. x ,  z
>.  ->  |^| |^| y  e.  _V )
1312adantr 270 . . . 4  |-  ( ( y  =  <. x ,  z >.  /\  (
x  e.  A  /\  z  e.  { B } ) )  ->  |^| |^| y  e.  _V )
1413exlimivv 1819 . . 3  |-  ( E. x E. z ( y  =  <. x ,  z >.  /\  (
x  e.  A  /\  z  e.  { B } ) )  ->  |^| |^| y  e.  _V )
155, 14sylbi 119 . 2  |-  ( y  e.  ( A  X.  { B } )  ->  |^| |^| y  e.  _V )
168, 2opex 4012 . . 3  |-  <. x ,  B >.  e.  _V
1716a1i 9 . 2  |-  ( x  e.  A  ->  <. x ,  B >.  e.  _V )
18 eqvisset 2618 . . . . 5  |-  ( x  =  |^| |^| y  ->  |^| |^| y  e.  _V )
19 ancom 262 . . . . . . . . . . 11  |-  ( ( ( y  =  <. x ,  z >.  /\  x  e.  A )  /\  z  e.  { B } )  <-> 
( z  e.  { B }  /\  (
y  =  <. x ,  z >.  /\  x  e.  A ) ) )
20 anass 393 . . . . . . . . . . 11  |-  ( ( ( y  =  <. x ,  z >.  /\  x  e.  A )  /\  z  e.  { B } )  <-> 
( y  =  <. x ,  z >.  /\  (
x  e.  A  /\  z  e.  { B } ) ) )
21 velsn 3433 . . . . . . . . . . . 12  |-  ( z  e.  { B }  <->  z  =  B )
2221anbi1i 446 . . . . . . . . . . 11  |-  ( ( z  e.  { B }  /\  ( y  = 
<. x ,  z >.  /\  x  e.  A
) )  <->  ( z  =  B  /\  (
y  =  <. x ,  z >.  /\  x  e.  A ) ) )
2319, 20, 223bitr3i 208 . . . . . . . . . 10  |-  ( ( y  =  <. x ,  z >.  /\  (
x  e.  A  /\  z  e.  { B } ) )  <->  ( z  =  B  /\  (
y  =  <. x ,  z >.  /\  x  e.  A ) ) )
2423exbii 1537 . . . . . . . . 9  |-  ( E. z ( y  = 
<. x ,  z >.  /\  ( x  e.  A  /\  z  e.  { B } ) )  <->  E. z
( z  =  B  /\  ( y  = 
<. x ,  z >.  /\  x  e.  A
) ) )
25 opeq2 3591 . . . . . . . . . . . 12  |-  ( z  =  B  ->  <. x ,  z >.  =  <. x ,  B >. )
2625eqeq2d 2094 . . . . . . . . . . 11  |-  ( z  =  B  ->  (
y  =  <. x ,  z >.  <->  y  =  <. x ,  B >. ) )
2726anbi1d 453 . . . . . . . . . 10  |-  ( z  =  B  ->  (
( y  =  <. x ,  z >.  /\  x  e.  A )  <->  ( y  =  <. x ,  B >.  /\  x  e.  A
) ) )
282, 27ceqsexv 2647 . . . . . . . . 9  |-  ( E. z ( z  =  B  /\  ( y  =  <. x ,  z
>.  /\  x  e.  A
) )  <->  ( y  =  <. x ,  B >.  /\  x  e.  A
) )
29 inteq 3659 . . . . . . . . . . . . . 14  |-  ( y  =  <. x ,  B >.  ->  |^| y  =  |^| <.
x ,  B >. )
3029inteqd 3661 . . . . . . . . . . . . 13  |-  ( y  =  <. x ,  B >.  ->  |^| |^| y  =  |^| |^|
<. x ,  B >. )
318, 2op1stb 4255 . . . . . . . . . . . . 13  |-  |^| |^| <. x ,  B >.  =  x
3230, 31syl6req 2132 . . . . . . . . . . . 12  |-  ( y  =  <. x ,  B >.  ->  x  =  |^| |^| y )
3332pm4.71ri 384 . . . . . . . . . . 11  |-  ( y  =  <. x ,  B >.  <-> 
( x  =  |^| |^| y  /\  y  = 
<. x ,  B >. ) )
3433anbi1i 446 . . . . . . . . . 10  |-  ( ( y  =  <. x ,  B >.  /\  x  e.  A )  <->  ( (
x  =  |^| |^| y  /\  y  =  <. x ,  B >. )  /\  x  e.  A
) )
35 anass 393 . . . . . . . . . 10  |-  ( ( ( x  =  |^| |^| y  /\  y  = 
<. x ,  B >. )  /\  x  e.  A
)  <->  ( x  = 
|^| |^| y  /\  (
y  =  <. x ,  B >.  /\  x  e.  A ) ) )
3634, 35bitri 182 . . . . . . . . 9  |-  ( ( y  =  <. x ,  B >.  /\  x  e.  A )  <->  ( x  =  |^| |^| y  /\  (
y  =  <. x ,  B >.  /\  x  e.  A ) ) )
3724, 28, 363bitri 204 . . . . . . . 8  |-  ( E. z ( y  = 
<. x ,  z >.  /\  ( x  e.  A  /\  z  e.  { B } ) )  <->  ( x  =  |^| |^| y  /\  (
y  =  <. x ,  B >.  /\  x  e.  A ) ) )
3837exbii 1537 . . . . . . 7  |-  ( E. x E. z ( y  =  <. x ,  z >.  /\  (
x  e.  A  /\  z  e.  { B } ) )  <->  E. x
( x  =  |^| |^| y  /\  ( y  =  <. x ,  B >.  /\  x  e.  A
) ) )
395, 38bitri 182 . . . . . 6  |-  ( y  e.  ( A  X.  { B } )  <->  E. x
( x  =  |^| |^| y  /\  ( y  =  <. x ,  B >.  /\  x  e.  A
) ) )
40 opeq1 3590 . . . . . . . . 9  |-  ( x  =  |^| |^| y  -> 
<. x ,  B >.  = 
<. |^| |^| y ,  B >. )
4140eqeq2d 2094 . . . . . . . 8  |-  ( x  =  |^| |^| y  ->  ( y  =  <. x ,  B >.  <->  y  =  <. |^| |^| y ,  B >. ) )
42 eleq1 2145 . . . . . . . 8  |-  ( x  =  |^| |^| y  ->  ( x  e.  A  <->  |^|
|^| y  e.  A
) )
4341, 42anbi12d 457 . . . . . . 7  |-  ( x  =  |^| |^| y  ->  ( ( y  = 
<. x ,  B >.  /\  x  e.  A )  <-> 
( y  =  <. |^|
|^| y ,  B >.  /\  |^| |^| y  e.  A
) ) )
4443ceqsexgv 2732 . . . . . 6  |-  ( |^| |^| y  e.  _V  ->  ( E. x ( x  =  |^| |^| y  /\  ( y  =  <. x ,  B >.  /\  x  e.  A ) )  <->  ( y  =  <. |^| |^| y ,  B >.  /\  |^| |^| y  e.  A
) ) )
4539, 44syl5bb 190 . . . . 5  |-  ( |^| |^| y  e.  _V  ->  ( y  e.  ( A  X.  { B }
)  <->  ( y  = 
<. |^| |^| y ,  B >.  /\  |^| |^| y  e.  A
) ) )
4618, 45syl 14 . . . 4  |-  ( x  =  |^| |^| y  ->  ( y  e.  ( A  X.  { B } )  <->  ( y  =  <. |^| |^| y ,  B >.  /\  |^| |^| y  e.  A
) ) )
4746pm5.32ri 443 . . 3  |-  ( ( y  e.  ( A  X.  { B }
)  /\  x  =  |^| |^| y )  <->  ( (
y  =  <. |^| |^| y ,  B >.  /\  |^| |^| y  e.  A )  /\  x  =  |^| |^| y ) )
4832adantr 270 . . . . 5  |-  ( ( y  =  <. x ,  B >.  /\  x  e.  A )  ->  x  =  |^| |^| y )
4948pm4.71i 383 . . . 4  |-  ( ( y  =  <. x ,  B >.  /\  x  e.  A )  <->  ( (
y  =  <. x ,  B >.  /\  x  e.  A )  /\  x  =  |^| |^| y ) )
5043pm5.32ri 443 . . . 4  |-  ( ( ( y  =  <. x ,  B >.  /\  x  e.  A )  /\  x  =  |^| |^| y )  <->  ( (
y  =  <. |^| |^| y ,  B >.  /\  |^| |^| y  e.  A )  /\  x  =  |^| |^| y ) )
5149, 50bitr2i 183 . . 3  |-  ( ( ( y  =  <. |^|
|^| y ,  B >.  /\  |^| |^| y  e.  A
)  /\  x  =  |^| |^| y )  <->  ( y  =  <. x ,  B >.  /\  x  e.  A
) )
52 ancom 262 . . 3  |-  ( ( y  =  <. x ,  B >.  /\  x  e.  A )  <->  ( x  e.  A  /\  y  =  <. x ,  B >. ) )
5347, 51, 523bitri 204 . 2  |-  ( ( y  e.  ( A  X.  { B }
)  /\  x  =  |^| |^| y )  <->  ( x  e.  A  /\  y  =  <. x ,  B >. ) )
544, 1, 15, 17, 53en2i 6338 1  |-  ( A  X.  { B }
)  ~~  A
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434   _Vcvv 2610   {csn 3416   <.cop 3419   |^|cint 3656   class class class wbr 3805    X. cxp 4389    ~~ cen 6306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-en 6309
This theorem is referenced by:  xpsneng  6387  endisj  6389
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