Theorem en2prd 6987

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 4318	. . . . 5 |- ( ( A e. V /\ C e. X ) -> <. A , C >. e. _V )
4	1, 2, 3	syl2anc 411	. . . 4 |- ( ph -> <. A , C >. e. _V )
5		en2prd.2	. . . . 5 |- ( ph -> B e. W )
6		en2prd.4	. . . . 5 |- ( ph -> D e. Y )
7		opexg 4318	. . . . 5 |- ( ( B e. W /\ D e. Y ) -> <. B , D >. e. _V )
8	5, 6, 7	syl2anc 411	. . . 4 |- ( ph -> <. B , D >. e. _V )
9		prexg 4299	. . . 4 |- ( ( <. A , C >. e. _V /\ <. B , D >. e. _V ) -> { <. A , C >. , <. B , D >. } e. _V )
10	4, 8, 9	syl2anc 411	. . 3 |- ( ph -> { <. A , C >. , <. B , D >. } e. _V )
11		en2prd.5	. . . 4 |- ( ph -> A =/= B )
12		en2prd.6	. . . 4 |- ( ph -> C =/= D )
13		f1oprg 5625	. . . . 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 1272	. . . 4 |- ( ph -> ( ( A =/= B /\ C =/= D ) -> { <. A , C >. , <. B , D >. } : { A , B } -1-1-onto-> { C , D } ) )
15	11, 12, 14	mp2and 433	. . 3 |- ( ph -> { <. A , C >. , <. B , D >. } : { A , B } -1-1-onto-> { C , D } )
16		f1oeq1 5568	. . 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 2949	. 2 |- ( ph -> E. f f : { A , B } -1-1-onto-> { C , D } )
18		prexg 4299	. . . 4 |- ( ( A e. V /\ B e. W ) -> { A , B } e. _V )
19	1, 5, 18	syl2anc 411	. . 3 |- ( ph -> { A , B } e. _V )
20		prexg 4299	. . . 4 |- ( ( C e. X /\ D e. Y ) -> { C , D } e. _V )
21	2, 6, 20	syl2anc 411	. . 3 |- ( ph -> { C , D } e. _V )
22		breng 6911	. . 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 411	. 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 1538   e. wcel 2200   =/= wne 2400   _Vcvv 2800   {cpr 3668   <.cop 3670   class class class wbr 4086   -1-1-onto->wf1o 5323   ~~ cen 6902

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-en 6905

This theorem is referenced by:  rex2dom  6991