Theorem carden 5981

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 5856).

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 3342	. . . . 5 |- ((card` A) = (card` B) -> (A ~~ (card` A) <-> A ~~ (card` B)))
2		cardid 5977	. . . . . 6 |- (card` B) ~~ B
3		entr 5473	. . . . . 6 |- ((A ~~ (card` B) /\ (card` B) ~~ B) -> A ~~ B)
4	2, 3	mpan2 760	. . . . 5 |- (A ~~ (card` B) -> A ~~ B)
5	1, 4	syl6bi 231	. . . 4 |- ((card` A) = (card` B) -> (A ~~ (card` A) -> A ~~ B))
6		cardid 5977	. . . . 5 |- (card` A) ~~ A
7		ensymg 5470	. . . . 5 |- (A e. C -> ((card` A) ~~ A -> A ~~ (card` A)))
8	6, 7	mpi 55	. . . 4 |- (A e. C -> A ~~ (card` A))
9	5, 8	syl5com 63	. . 3 |- (A e. C -> ((card` A) = (card` B) -> A ~~ B))
10	9	adantr 425	. 2 |- ((A e. C /\ B e. D) -> ((card` A) = (card` B) -> A ~~ B))
11		ensymg 5470	. . . . . 6 |- (B e. D -> (A ~~ B -> B ~~ A))
12		entr 5473	. . . . . . . 8 |- (((card` B) ~~ B /\ B ~~ A) -> (card` B) ~~ A)
13	2, 12	mpan 759	. . . . . . 7 |- (B ~~ A -> (card` B) ~~ A)
14		cardne 5980	. . . . . . . . 9 |- ((card` B) e. (card` A) -> -. (card` B) ~~ A)
15	14	con2i 113	. . . . . . . 8 |- ((card` B) ~~ A -> -. (card` B) e. (card` A))
16		cardon 5976	. . . . . . . . 9 |- (card` A) e. On
17		cardon 5976	. . . . . . . . 9 |- (card` B) e. On
18		ontri1 3695	. . . . . . . . 9 |- (((card` A) e. On /\ (card` B) e. On) -> ((card` A) C_ (card` B) <-> -. (card` B) e. (card` A)))
19	16, 17, 18	mp2an 761	. . . . . . . 8 |- ((card` A) C_ (card` B) <-> -. (card` B) e. (card` A))
20	15, 19	sylibr 217	. . . . . . 7 |- ((card` B) ~~ A -> (card` A) C_ (card` B))
21	13, 20	syl 12	. . . . . 6 |- (B ~~ A -> (card` A) C_ (card` B))
22	11, 21	syl6 25	. . . . 5 |- (B e. D -> (A ~~ B -> (card` A) C_ (card` B)))
23		entr 5473	. . . . . . . 8 |- (((card` A) ~~ A /\ A ~~ B) -> (card` A) ~~ B)
24	6, 23	mpan 759	. . . . . . 7 |- (A ~~ B -> (card` A) ~~ B)
25		cardne 5980	. . . . . . . . 9 |- ((card` A) e. (card` B) -> -. (card` A) ~~ B)
26	25	con2i 113	. . . . . . . 8 |- ((card` A) ~~ B -> -. (card` A) e. (card` B))
27		ontri1 3695	. . . . . . . . 9 |- (((card` B) e. On /\ (card` A) e. On) -> ((card` B) C_ (card` A) <-> -. (card` A) e. (card` B)))
28	17, 16, 27	mp2an 761	. . . . . . . 8 |- ((card` B) C_ (card` A) <-> -. (card` A) e. (card` B))
29	26, 28	sylibr 217	. . . . . . 7 |- ((card` A) ~~ B -> (card` B) C_ (card` A))
30	24, 29	syl 12	. . . . . 6 |- (A ~~ B -> (card` B) C_ (card` A))
31	30	a1i 8	. . . . 5 |- (B e. D -> (A ~~ B -> (card` B) C_ (card` A)))
32	22, 31	jcad 661	. . . 4 |- (B e. D -> (A ~~ B -> ((card` A) C_ (card` B) /\ (card` B) C_ (card` A))))
33		eqss 2631	. . . 4 |- ((card` A) = (card` B) <-> ((card` A) C_ (card` B) /\ (card` B) C_ (card` A)))
34	32, 33	syl6ibr 230	. . 3 |- (B e. D -> (A ~~ B -> (card` A) = (card` B)))
35	34	adantl 424	. 2 |- ((A e. C /\ B e. D) -> (A ~~ B -> (card` A) = (card` B)))
36	10, 35	impbid 574	1 |- ((A e. C /\ B e. D) -> ((card` A) = (card` B) <-> A ~~ B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   C_ wss 2593   class class class wbr 3338  Oncon0 3657  ` cfv 3998   ~~ cen 5423  cardccrd 5859

This theorem is referenced by:  cardeq0 5982  card1 5983  carddom 5987  cardsdom 5988  cardsucinf 5993  cardidm 6001  cfom 6064  nnacda 6088  nnaun 6089  cardfz 7719  hashen 8233  cardennn 10171  ficard 10176  carinttar2 15280

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-er 5318  df-en 5427  df-card 5862

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