Theorem carden2bex 7358

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 6998	. . . . 5 |- ( A ~~ B -> ( y ~~ A <-> y ~~ B ) )
2	1	rabbidv 2788	. . . 4 |- ( A ~~ B -> { y e. On | y ~~ A } = { y e. On | y ~~ B } )
3	2	inteqd 3927	. . 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 7353	. . 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 6934	. . . . . 6 |- ( ( x ~~ A /\ A ~~ B ) -> x ~~ B )
8	7	expcom 116	. . . . 5 |- ( A ~~ B -> ( x ~~ A -> x ~~ B ) )
9	8	reximdv 2631	. . . 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 7353	. . 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 2272	1 |- ( ( A ~~ B /\ E. x e. On x ~~ A ) -> ( card ` A ) = ( card ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   E.wrex 2509   {crab 2512   |^|cint 3922   class class class wbr 4082   Oncon0 4453   ` cfv 5317   ~~ cen 6883   cardccrd 7345

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 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-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  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-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-er 6678  df-en 6886  df-card 7347

This theorem is referenced by: (None)