Theorem carddom 3642

Description: Two sets have the dominance relationship iff their cardinalities have the subset relationship. Equation (i) of [Quine] p. 232.

Assertion

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

Proof of Theorem carddom

Step	Hyp	Ref	Expression
1		carddomi 3641	. . 3 |- (A e. C -> ((card` A) (_ (card` B) -> A ~<_ B))
2	1	adantr 306	. 2 |- ((A e. C /\ B e. D) -> ((card` A) (_ (card` B) -> A ~<_ B))
3		carddomi 3641	. . . . . . . . 9 |- (B e. D -> ((card` B) (_ (card` A) -> B ~<_ A))
4		cardon 3634	. . . . . . . . . 10 |- (card` A) e. On
5	4	onelss 2348	. . . . . . . . 9 |- ((card` B) e. (card` A) -> (card` B) (_ (card` A))
6	3, 5	syl5 22	. . . . . . . 8 |- (B e. D -> ((card` B) e. (card` A) -> B ~<_ A))
7		domnsym 3365	. . . . . . . 8 |- (B ~<_ A -> -. A ~< B)
8	6, 7	syl6 23	. . . . . . 7 |- (B e. D -> ((card` B) e. (card` A) -> -. A ~< B))
9	8	con2d 83	. . . . . 6 |- (B e. D -> (A ~< B -> -. (card` B) e. (card` A)))
10		cardon 3634	. . . . . . 7 |- (card` B) e. On
11		ontri1 2232	. . . . . . 7 |- (((card` A) e. On /\ (card` B) e. On) -> ((card` A) (_ (card` B) <-> -. (card` B) e. (card` A)))
12	4, 10, 11	mp2an 520	. . . . . 6 |- ((card` A) (_ (card` B) <-> -. (card` B) e. (card` A))
13	9, 12	syl6ibr 186	. . . . 5 |- (B e. D -> (A ~< B -> (card` A) (_ (card` B)))
14	13	adantl 305	. . . 4 |- ((A e. C /\ B e. D) -> (A ~< B -> (card` A) (_ (card` B)))
15		carden 3638	. . . . 5 |- ((A e. C /\ B e. D) -> ((card` A) = (card` B) <-> A ~~ B))
16		eqimss 1548	. . . . 5 |- ((card` A) = (card` B) -> (card` A) (_ (card` B))
17	15, 16	syl6bir 188	. . . 4 |- ((A e. C /\ B e. D) -> (A ~~ B -> (card` A) (_ (card` B)))
18	14, 17	jaod 329	. . 3 |- ((A e. C /\ B e. D) -> ((A ~< B \/ A ~~ B) -> (card` A) (_ (card` B)))
19		brdom2 3292	. . 3 |- (A ~<_ B <-> (A ~< B \/ A ~~ B))
20	18, 19	syl5ib 181	. 2 |- ((A e. C /\ B e. D) -> (A ~<_ B -> (card` A) (_ (card` B)))
21	2, 20	impbid 397	1 |- ((A e. C /\ B e. D) -> ((card` A) (_ (card` B) <-> A ~<_ B))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196   = wceq 1091   e. wcel 1092   (_ wss 1487   class class class wbr 2054  Oncon0 2199  ` cfv 2422   ~~ cen 3271   ~<_ cdom 3272   ~< csdm 3273  cardccrd 3620

This theorem is referenced by:  cardsdom 3643  domtri 3644  carduni 3664  cardprc 3667  cardaleph 3690

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623

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