Theorem carden 4831

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 4726).

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 2623	. . . . 5 |- ((card` A) = (card` B) -> (A ~~ (card` A) <-> A ~~ (card` B)))
2		cardid 4828	. . . . . 6 |- (card` B) ~~ B
3		entrt 4414	. . . . . 6 |- ((A ~~ (card` B) /\ (card` B) ~~ B) -> A ~~ B)
4	2, 3	mpan2 696	. . . . 5 |- (A ~~ (card` B) -> A ~~ B)
5	1, 4	syl6bi 214	. . . 4 |- ((card` A) = (card` B) -> (A ~~ (card` A) -> A ~~ B))
6		cardid 4828	. . . . 5 |- (card` A) ~~ A
7		ensymg 4411	. . . . 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 4411	. . . . . 6 |- (B e. D -> (A ~~ B -> B ~~ A))
12		entrt 4414	. . . . . . . 8 |- (((card` B) ~~ B /\ B ~~ A) -> (card` B) ~~ A)
13	2, 12	mpan 695	. . . . . . 7 |- (B ~~ A -> (card` B) ~~ A)
14		cardne 4830	. . . . . . . . 9 |- ((card` B) e. (card` A) -> -. (card` B) ~~ A)
15	14	con2i 97	. . . . . . . 8 |- ((card` B) ~~ A -> -. (card` B) e. (card` A))
16		cardon 4827	. . . . . . . . 9 |- (card` A) e. On
17		cardon 4827	. . . . . . . . 9 |- (card` B) e. On
18		ontri1 2981	. . . . . . . . 9 |- (((card` A) e. On /\ (card` B) e. On) -> ((card` A) (_ (card` B) <-> -. (card` B) e. (card` A)))
19	16, 17, 18	mp2an 697	. . . . . . . 8 |- ((card` A) (_ (card` B) <-> -. (card` B) e. (card` A))
20	15, 19	sylibr 200	. . . . . . 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		entrt 4414	. . . . . . . 8 |- (((card` A) ~~ A /\ A ~~ B) -> (card` A) ~~ B)
24	6, 23	mpan 695	. . . . . . 7 |- (A ~~ B -> (card` A) ~~ B)
25		cardne 4830	. . . . . . . . 9 |- ((card` A) e. (card` B) -> -. (card` A) ~~ B)
26	25	con2i 97	. . . . . . . 8 |- ((card` A) ~~ B -> -. (card` A) e. (card` B))
27		ontri1 2981	. . . . . . . . 9 |- (((card` B) e. On /\ (card` A) e. On) -> ((card` B) (_ (card` A) <-> -. (card` A) e. (card` B)))
28	17, 16, 27	mp2an 697	. . . . . . . 8 |- ((card` B) (_ (card` A) <-> -. (card` A) e. (card` B))
29	26, 28	sylibr 200	. . . . . . 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 600	. . . 4 |- (B e. D -> (A ~~ B -> ((card` A) (_ (card` B) /\ (card` B) (_ (card` A))))
33		eqss 2077	. . . 4 |- ((card` A) = (card` B) <-> ((card` A) (_ (card` B) /\ (card` B) (_ (card` A)))
34	32, 33	syl6ibr 213	. . 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 516	1 |- ((A e. C /\ B e. D) -> ((card` A) = (card` B) <-> A ~~ B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958   (_ wss 2047   class class class wbr 2619  Oncon0 2948  ` cfv 3182   ~~ cen 4364  cardccrd 4813

This theorem is referenced by:  cardeq0 4832  card1 4833  carddom 4836  cardsdom 4837  cardidm 4849  cfom 4916

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-er 4261  df-en 4368  df-card 4816

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