Theorem carden2bex 7296

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 6937	. . . . 5 |- ( A ~~ B -> ( y ~~ A <-> y ~~ B ) )
2	1	rabbidv 2760	. . . 4 |- ( A ~~ B -> { y e. On | y ~~ A } = { y e. On | y ~~ B } )
3	2	inteqd 3889	. . 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 7291	. . 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 6875	. . . . . 6 |- ( ( x ~~ A /\ A ~~ B ) -> x ~~ B )
8	7	expcom 116	. . . . 5 |- ( A ~~ B -> ( x ~~ A -> x ~~ B ) )
9	8	reximdv 2606	. . . 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 7291	. . 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 2247	1 |- ( ( A ~~ B /\ E. x e. On x ~~ A ) -> ( card ` A ) = ( card ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   E.wrex 2484   {crab 2487   |^|cint 3884   class class class wbr 4043   Oncon0 4409   ` cfv 5270   ~~ cen 6824   cardccrd 7283

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-er 6619  df-en 6827  df-card 7285

This theorem is referenced by: (None)