Theorem carden 4979

Description: Two sets are equinumerous iff their cardinal numbers are equal. This important theorem expresses the essential concept behind "cardinality" or "size." This theorem appears as Proposition 10.10 of [TakeutiZaring] p. 85, Theorem 7P of [Enderton] p. 197, and Theorem 9 of [Suppes] p. 242 (among others). The Axiom of Choice is required for its proof.

The theory of cardinality can also be developed without AC by introducing "card" as a primitive notion and stating this theorem as an axiom, as is done with the axiom for cardinal numbers in [Suppes] p. 111. Finally, if we allow the Axiom of Regularity, we can avoid AC by defining the cardinal number of a set as the set of all sets equinumerous to it and having least possible rank (see karden 4872).

Assertion

Ref	Expression
carden	|- ((A e. C /\ B e. D) -> ((card` A) = (card` B) <-> A ~~ B))

Proof of Theorem carden

Step	Hyp	Ref	Expression
1		breq2 2696	. . . . 5 |- ((card` A) = (card` B) -> (A ~~ (card` A) <-> A ~~ (card` B)))
2		cardid 4975	. . . . . 6 |- (card` B) ~~ B
3		entr 4555	. . . . . 6 |- ((A ~~ (card` B) /\ (card` B) ~~ B) -> A ~~ B)
4	2, 3	mpan2 700	. . . . 5 |- (A ~~ (card` B) -> A ~~ B)
5	1, 4	syl6bi 212	. . . 4 |- ((card` A) = (card` B) -> (A ~~ (card` A) -> A ~~ B))
6		cardid 4975	. . . . 5 |- (card` A) ~~ A
7		ensymg 4552	. . . . 5 |- (A e. C -> ((card` A) ~~ A -> A ~~ (card` A)))
8	6, 7	mpi 44	. . . 4 |- (A e. C -> A ~~ (card` A))
9	5, 8	syl5com 52	. . 3 |- (A e. C -> ((card` A) = (card` B) -> A ~~ B))
10	9	adantr 389	. 2 |- ((A e. C /\ B e. D) -> ((card` A) = (card` B) -> A ~~ B))
11		ensymg 4552	. . . . . 6 |- (B e. D -> (A ~~ B -> B ~~ A))
12		entr 4555	. . . . . . . 8 |- (((card` B) ~~ B /\ B ~~ A) -> (card` B) ~~ A)
13	2, 12	mpan 699	. . . . . . 7 |- (B ~~ A -> (card` B) ~~ A)
14		cardne 4978	. . . . . . . . 9 |- ((card` B) e. (card` A) -> -. (card` B) ~~ A)
15	14	con2i 97	. . . . . . . 8 |- ((card` B) ~~ A -> -. (card` B) e. (card` A))
16		cardon 4974	. . . . . . . . 9 |- (card` A) e. On
17		cardon 4974	. . . . . . . . 9 |- (card` B) e. On
18		ontri1 3009	. . . . . . . . 9 |- (((card` A) e. On /\ (card` B) e. On) -> ((card` A) (_ (card` B) <-> -. (card` B) e. (card` A)))
19	16, 17, 18	mp2an 701	. . . . . . . 8 |- ((card` A) (_ (card` B) <-> -. (card` B) e. (card` A))
20	15, 19	sylibr 198	. . . . . . 7 |- ((card` B) ~~ A -> (card` A) (_ (card` B))
21	13, 20	syl 10	. . . . . 6 |- (B ~~ A -> (card` A) (_ (card` B))
22	11, 21	syl6 22	. . . . 5 |- (B e. D -> (A ~~ B -> (card` A) (_ (card` B)))
23		entr 4555	. . . . . . . 8 |- (((card` A) ~~ A /\ A ~~ B) -> (card` A) ~~ B)
24	6, 23	mpan 699	. . . . . . 7 |- (A ~~ B -> (card` A) ~~ B)
25		cardne 4978	. . . . . . . . 9 |- ((card` A) e. (card` B) -> -. (card` A) ~~ B)
26	25	con2i 97	. . . . . . . 8 |- ((card` A) ~~ B -> -. (card` A) e. (card` B))
27		ontri1 3009	. . . . . . . . 9 |- (((card` B) e. On /\ (card` A) e. On) -> ((card` B) (_ (card` A) <-> -. (card` A) e. (card` B)))
28	17, 16, 27	mp2an 701	. . . . . . . 8 |- ((card` B) (_ (card` A) <-> -. (card` A) e. (card` B))
29	26, 28	sylibr 198	. . . . . . 7 |- ((card` A) ~~ B -> (card` B) (_ (card` A))
30	24, 29	syl 10	. . . . . 6 |- (A ~~ B -> (card` B) (_ (card` A))
31	30	a1i 8	. . . . 5 |- (B e. D -> (A ~~ B -> (card` B) (_ (card` A)))
32	22, 31	jcad 603	. . . 4 |- (B e. D -> (A ~~ B -> ((card` A) (_ (card` B) /\ (card` B) (_ (card` A))))
33		eqss 2129	. . . 4 |- ((card` A) = (card` B) <-> ((card` A) (_ (card` B) /\ (card` B) (_ (card` A)))
34	32, 33	syl6ibr 211	. . 3 |- (B e. D -> (A ~~ B -> (card` A) = (card` B)))
35	34	adantl 388	. 2 |- ((A e. C /\ B e. D) -> (A ~~ B -> (card` A) = (card` B)))
36	10, 35	impbid 519	1 |- ((A e. C /\ B e. D) -> ((card` A) = (card` B) <-> A ~~ B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   /\ wa 221   = wceq 992   e. wcel 994   (_ wss 2099   class class class wbr 2692  Oncon0 2975  ` cfv 3263   ~~ cen 4505  cardccrd 4959

This theorem is referenced by:  cardeq0 4980  card1 4981  carddom 4985  cardsdom 4986  cardsucinf 4991  cardidm 4999  cfom 5066  nnacda 5090  nnaun 5091  cardfz 6671  cardennn 11832  ficard 11834

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-ac 4890

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-suc 2981  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-er 4401  df-en 4509  df-card 4962

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