Theorem carden2bex 6458

Description: If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.)

Assertion

Ref	Expression
carden2bex	|- ( ( A ~~ B /\ E. x e. On x ~~ A ) -> ( card ` A ) = ( card ` B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem carden2bex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		enen2 6335	. . . . 5 |- ( A ~~ B -> ( y ~~ A <-> y ~~ B ) )
2	1	rabbidv 2593	. . . 4 |- ( A ~~ B -> { y e. On | y ~~ A } = { y e. On | y ~~ B } )
3	2	inteqd 3641	. . 3 |- ( A ~~ B -> |^| { y e. On | y ~~ A } = |^| { y e. On | y ~~ B } )
4	3	adantr 270	. 2 |- ( ( A ~~ B /\ E. x e. On x ~~ A ) -> |^| { y e. On | y ~~ A } = |^| { y e. On | y ~~ B } )
5		cardval3ex 6454	. . 3 |- ( E. x e. On x ~~ A -> ( card ` A ) = |^| { y e. On | y ~~ A } )
6	5	adantl 271	. 2 |- ( ( A ~~ B /\ E. x e. On x ~~ A ) -> ( card ` A ) = |^| { y e. On | y ~~ A } )
7		entr 6287	. . . . . 6 |- ( ( x ~~ A /\ A ~~ B ) -> x ~~ B )
8	7	expcom 114	. . . . 5 |- ( A ~~ B -> ( x ~~ A -> x ~~ B ) )
9	8	reximdv 2462	. . . 4 |- ( A ~~ B -> ( E. x e. On x ~~ A -> E. x e. On x ~~ B ) )
10	9	imp 122	. . 3 |- ( ( A ~~ B /\ E. x e. On x ~~ A ) -> E. x e. On x ~~ B )
11		cardval3ex 6454	. . 3 |- ( E. x e. On x ~~ B -> ( card ` B ) = |^| { y e. On | y ~~ B } )
12	10, 11	syl 14	. 2 |- ( ( A ~~ B /\ E. x e. On x ~~ A ) -> ( card ` B ) = |^| { y e. On | y ~~ B } )
13	4, 6, 12	3eqtr4d 2123	1 |- ( ( A ~~ B /\ E. x e. On x ~~ A ) -> ( card ` A ) = ( card ` B ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   E.wrex 2349   {crab 2352   |^|cint 3636   class class class wbr 3785   Oncon0 4118   ` cfv 4922   ~~ cen 6242   cardccrd 6448

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-er 6129  df-en 6245  df-card 6449

This theorem is referenced by: (None)