Theorem en2prd 7096

Description: Two proper unordered pairs are equinumerous. (Contributed by BTernaryTau, 23-Dec-2024.)

Hypotheses

Ref	Expression
en2prd.1	|- ( ph -> A e. V )
en2prd.2	|- ( ph -> B e. W )
en2prd.3	|- ( ph -> C e. X )
en2prd.4	|- ( ph -> D e. Y )
en2prd.5	|- ( ph -> A =/= B )
en2prd.6	|- ( ph -> C =/= D )

Assertion

Ref	Expression
en2prd	|- ( ph -> { A , B } ~~ { C , D } )

Proof of Theorem en2prd

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		en2prd.1	. . . . 5 |- ( ph -> A e. V )
2		en2prd.3	. . . . 5 |- ( ph -> C e. X )
3		opexg 4363	. . . . 5 |- ( ( A e. V /\ C e. X ) -> <. A , C >. e. _V )
4	1, 2, 3	syl2anc 415	. . . 4 |- ( ph -> <. A , C >. e. _V )
5		en2prd.2	. . . . 5 |- ( ph -> B e. W )
6		en2prd.4	. . . . 5 |- ( ph -> D e. Y )
7		opexg 4363	. . . . 5 |- ( ( B e. W /\ D e. Y ) -> <. B , D >. e. _V )
8	5, 6, 7	syl2anc 415	. . . 4 |- ( ph -> <. B , D >. e. _V )
9		prexg 4344	. . . 4 |- ( ( <. A , C >. e. _V /\ <. B , D >. e. _V ) -> { <. A , C >. , <. B , D >. } e. _V )
10	4, 8, 9	syl2anc 415	. . 3 |- ( ph -> { <. A , C >. , <. B , D >. } e. _V )
11		en2prd.5	. . . 4 |- ( ph -> A =/= B )
12		en2prd.6	. . . 4 |- ( ph -> C =/= D )
13		f1oprg 5680	. . . . 5 |- ( ( ( A e. V /\ C e. X ) /\ ( B e. W /\ D e. Y ) ) -> ( ( A =/= B /\ C =/= D ) -> { <. A , C >. , <. B , D >. } : { A , B } -1-1-onto-> { C , D } ) )
14	1, 2, 5, 6, 13	syl22anc 1279	. . . 4 |- ( ph -> ( ( A =/= B /\ C =/= D ) -> { <. A , C >. , <. B , D >. } : { A , B } -1-1-onto-> { C , D } ) )
15	11, 12, 14	mp2and 437	. . 3 |- ( ph -> { <. A , C >. , <. B , D >. } : { A , B } -1-1-onto-> { C , D } )
16		f1oeq1 5622	. . 3 |- ( f = { <. A , C >. , <. B , D >. } -> ( f : { A , B } -1-1-onto-> { C , D } <-> { <. A , C >. , <. B , D >. } : { A , B } -1-1-onto-> { C , D } ) )
17	10, 15, 16	elabd 2971	. 2 |- ( ph -> E. f f : { A , B } -1-1-onto-> { C , D } )
18		prexg 4344	. . . 4 |- ( ( A e. V /\ B e. W ) -> { A , B } e. _V )
19	1, 5, 18	syl2anc 415	. . 3 |- ( ph -> { A , B } e. _V )
20		prexg 4344	. . . 4 |- ( ( C e. X /\ D e. Y ) -> { C , D } e. _V )
21	2, 6, 20	syl2anc 415	. . 3 |- ( ph -> { C , D } e. _V )
22		breng 7019	. . 3 |- ( ( { A , B } e. _V /\ { C , D } e. _V ) -> ( { A , B } ~~ { C , D } <-> E. f f : { A , B } -1-1-onto-> { C , D } ) )
23	19, 21, 22	syl2anc 415	. 2 |- ( ph -> ( { A , B } ~~ { C , D } <-> E. f f : { A , B } -1-1-onto-> { C , D } ) )
24	17, 23	mpbird 167	1 |- ( ph -> { A , B } ~~ { C , D } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1545   e. wcel 2209   =/= wne 2420   _Vcvv 2821   {cpr 3706   <.cop 3708   class class class wbr 4125   -1-1-onto->wf1o 5371   ~~ cen 7010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-en 7013

This theorem is referenced by:  rex2dom  7100