Theorem carden2bex 7045

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 6735	. . . . 5 |- ( A ~~ B -> ( y ~~ A <-> y ~~ B ) )
2	1	rabbidv 2675	. . . 4 |- ( A ~~ B -> { y e. On | y ~~ A } = { y e. On | y ~~ B } )
3	2	inteqd 3776	. . 3 |- ( A ~~ B -> |^| { y e. On | y ~~ A } = |^| { y e. On | y ~~ B } )
4	3	adantr 274	. 2 |- ( ( A ~~ B /\ E. x e. On x ~~ A ) -> |^| { y e. On | y ~~ A } = |^| { y e. On | y ~~ B } )
5		cardval3ex 7041	. . 3 |- ( E. x e. On x ~~ A -> ( card ` A ) = |^| { y e. On | y ~~ A } )
6	5	adantl 275	. 2 |- ( ( A ~~ B /\ E. x e. On x ~~ A ) -> ( card ` A ) = |^| { y e. On | y ~~ A } )
7		entr 6678	. . . . . 6 |- ( ( x ~~ A /\ A ~~ B ) -> x ~~ B )
8	7	expcom 115	. . . . 5 |- ( A ~~ B -> ( x ~~ A -> x ~~ B ) )
9	8	reximdv 2533	. . . 4 |- ( A ~~ B -> ( E. x e. On x ~~ A -> E. x e. On x ~~ B ) )
10	9	imp 123	. . 3 |- ( ( A ~~ B /\ E. x e. On x ~~ A ) -> E. x e. On x ~~ B )
11		cardval3ex 7041	. . 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 2182	1 |- ( ( A ~~ B /\ E. x e. On x ~~ A ) -> ( card ` A ) = ( card ` B ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   E.wrex 2417   {crab 2420   |^|cint 3771   class class class wbr 3929   Oncon0 4285   ` cfv 5123   ~~ cen 6632   cardccrd 7035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-er 6429  df-en 6635  df-card 7036

This theorem is referenced by: (None)