Theorem carden2bex 7525

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 7131	. . . . 5 |- ( A ~~ B -> ( y ~~ A <-> y ~~ B ) )
2	1	rabbidv 2810	. . . 4 |- ( A ~~ B -> { y e. On | y ~~ A } = { y e. On | y ~~ B } )
3	2	inteqd 3970	. . 3 |- ( A ~~ B -> |^| { y e. On | y ~~ A } = |^| { y e. On | y ~~ B } )
4	3	adantr 276	. 2 |- ( ( A ~~ B /\ E. x e. On x ~~ A ) -> |^| { y e. On | y ~~ A } = |^| { y e. On | y ~~ B } )
5		cardval3ex 7520	. . 3 |- ( E. x e. On x ~~ A -> ( card ` A ) = |^| { y e. On | y ~~ A } )
6	5	adantl 277	. 2 |- ( ( A ~~ B /\ E. x e. On x ~~ A ) -> ( card ` A ) = |^| { y e. On | y ~~ A } )
7		entr 7061	. . . . . 6 |- ( ( x ~~ A /\ A ~~ B ) -> x ~~ B )
8	7	expcom 116	. . . . 5 |- ( A ~~ B -> ( x ~~ A -> x ~~ B ) )
9	8	reximdv 2651	. . . 4 |- ( A ~~ B -> ( E. x e. On x ~~ A -> E. x e. On x ~~ B ) )
10	9	imp 124	. . 3 |- ( ( A ~~ B /\ E. x e. On x ~~ A ) -> E. x e. On x ~~ B )
11		cardval3ex 7520	. . 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 2281	1 |- ( ( A ~~ B /\ E. x e. On x ~~ A ) -> ( card ` A ) = ( card ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   E.wrex 2529   {crab 2532   |^|cint 3965   class class class wbr 4125   Oncon0 4503   ` cfv 5372   ~~ cen 7010   cardccrd 7512

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-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  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  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-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-er 6797  df-en 7013  df-card 7514

This theorem is referenced by: (None)