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Theorem op1stbg 4238
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by Jim Kingdon, 17-Dec-2018.)
Assertion
Ref Expression
op1stbg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| |^| <. A ,  B >.  =  A )

Proof of Theorem op1stbg
StepHypRef Expression
1 dfopg 3575 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
21inteqd 3648 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| <. A ,  B >.  =  |^| { { A } ,  { A ,  B } } )
3 elex 2583 . . . . . . . 8  |-  ( A  e.  V  ->  A  e.  _V )
4 snexgOLD 3963 . . . . . . . 8  |-  ( A  e.  _V  ->  { A }  e.  _V )
53, 4syl 14 . . . . . . 7  |-  ( A  e.  V  ->  { A }  e.  _V )
65adantr 265 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { A }  e.  _V )
7 elex 2583 . . . . . . 7  |-  ( B  e.  W  ->  B  e.  _V )
8 prexgOLD 3974 . . . . . . 7  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { A ,  B }  e.  _V )
93, 7, 8syl2an 277 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { A ,  B }  e.  _V )
10 intprg 3676 . . . . . 6  |-  ( ( { A }  e.  _V  /\  { A ,  B }  e.  _V )  ->  |^| { { A } ,  { A ,  B } }  =  ( { A }  i^i  { A ,  B }
) )
116, 9, 10syl2anc 397 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { { A } ,  { A ,  B } }  =  ( { A }  i^i  { A ,  B }
) )
12 snsspr1 3540 . . . . . 6  |-  { A }  C_  { A ,  B }
13 df-ss 2959 . . . . . 6  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  i^i  { A ,  B } )  =  { A } )
1412, 13mpbi 137 . . . . 5  |-  ( { A }  i^i  { A ,  B }
)  =  { A }
1511, 14syl6eq 2104 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { { A } ,  { A ,  B } }  =  { A } )
162, 15eqtrd 2088 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| <. A ,  B >.  =  { A }
)
1716inteqd 3648 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| |^| <. A ,  B >.  =  |^| { A } )
18 intsng 3677 . . 3  |-  ( A  e.  V  ->  |^| { A }  =  A )
1918adantr 265 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { A }  =  A )
2017, 19eqtrd 2088 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| |^| <. A ,  B >.  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259    e. wcel 1409   _Vcvv 2574    i^i cin 2944    C_ wss 2945   {csn 3403   {cpr 3404   <.cop 3406   |^|cint 3643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-int 3644
This theorem is referenced by:  elxp5  4837  fundmen  6317
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