Theorem carden 4814

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 4709).

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 2619	. . . . 5 |- ((card` A) = (card` B) -> (A ~~ (card` A) <-> A ~~ (card` B)))
2		cardid 4811	. . . . . 6 |- (card` B) ~~ B
3		entrt 4404	. . . . . 6 |- ((A ~~ (card` B) /\ (card` B) ~~ B) -> A ~~ B)
4	2, 3	mpan2 695	. . . . 5 |- (A ~~ (card` B) -> A ~~ B)
5	1, 4	syl6bi 214	. . . 4 |- ((card` A) = (card` B) -> (A ~~ (card` A) -> A ~~ B))
6		cardid 4811	. . . . 5 |- (card` A) ~~ A
7		ensymg 4401	. . . . 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 4401	. . . . . 6 |- (B e. D -> (A ~~ B -> B ~~ A))
12		entrt 4404	. . . . . . . 8 |- (((card` B) ~~ B /\ B ~~ A) -> (card` B) ~~ A)
13	2, 12	mpan 694	. . . . . . 7 |- (B ~~ A -> (card` B) ~~ A)
14		cardne 4813	. . . . . . . . 9 |- ((card` B) e. (card` A) -> -. (card` B) ~~ A)
15	14	con2i 97	. . . . . . . 8 |- ((card` B) ~~ A -> -. (card` B) e. (card` A))
16		cardon 4810	. . . . . . . . 9 |- (card` A) e. On
17		cardon 4810	. . . . . . . . 9 |- (card` B) e. On
18		ontri1 2977	. . . . . . . . 9 |- (((card` A) e. On /\ (card` B) e. On) -> ((card` A) (_ (card` B) <-> -. (card` B) e. (card` A)))
19	16, 17, 18	mp2an 696	. . . . . . . 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 4404	. . . . . . . 8 |- (((card` A) ~~ A /\ A ~~ B) -> (card` A) ~~ B)
24	6, 23	mpan 694	. . . . . . 7 |- (A ~~ B -> (card` A) ~~ B)
25		cardne 4813	. . . . . . . . 9 |- ((card` A) e. (card` B) -> -. (card` A) ~~ B)
26	25	con2i 97	. . . . . . . 8 |- ((card` A) ~~ B -> -. (card` A) e. (card` B))
27		ontri1 2977	. . . . . . . . 9 |- (((card` B) e. On /\ (card` A) e. On) -> ((card` B) (_ (card` A) <-> -. (card` A) e. (card` B)))
28	17, 16, 27	mp2an 696	. . . . . . . 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 599	. . . 4 |- (B e. D -> (A ~~ B -> ((card` A) (_ (card` B) /\ (card` B) (_ (card` A))))
33		eqss 2074	. . . 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 515	1 |- ((A e. C /\ B e. D) -> ((card` A) = (card` B) <-> A ~~ B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957   (_ wss 2044   class class class wbr 2615  Oncon0 2944  ` cfv 3178   ~~ cen 4357  cardccrd 4796

This theorem is referenced by:  cardeq0 4815  card1 4816  carddom 4819  cardsdom 4820  cardidm 4832  cfom 4899

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-ac 4727

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-suc 2950  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-er 4254  df-en 4360  df-card 4799

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