Theorem carddom 4819

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 4818	. . 3 |- (A e. C -> ((card` A) (_ (card` B) -> A ~<_ B))
2	1	adantr 389	. 2 |- ((A e. C /\ B e. D) -> ((card` A) (_ (card` B) -> A ~<_ B))
3		carddomi 4818	. . . . . . . . 9 |- (B e. D -> ((card` B) (_ (card` A) -> B ~<_ A))
4		cardon 4810	. . . . . . . . . 10 |- (card` A) e. On
5	4	onelss 3096	. . . . . . . . 9 |- ((card` B) e. (card` A) -> (card` B) (_ (card` A))
6	3, 5	syl5 21	. . . . . . . 8 |- (B e. D -> ((card` B) e. (card` A) -> B ~<_ A))
7		domnsym 4452	. . . . . . . 8 |- (B ~<_ A -> -. A ~< B)
8	6, 7	syl6 22	. . . . . . 7 |- (B e. D -> ((card` B) e. (card` A) -> -. A ~< B))
9	8	con2d 91	. . . . . 6 |- (B e. D -> (A ~< B -> -. (card` B) e. (card` A)))
10		cardon 4810	. . . . . . 7 |- (card` B) e. On
11		ontri1 2977	. . . . . . 7 |- (((card` A) e. On /\ (card` B) e. On) -> ((card` A) (_ (card` B) <-> -. (card` B) e. (card` A)))
12	4, 10, 11	mp2an 696	. . . . . 6 |- ((card` A) (_ (card` B) <-> -. (card` B) e. (card` A))
13	9, 12	syl6ibr 213	. . . . 5 |- (B e. D -> (A ~< B -> (card` A) (_ (card` B)))
14	13	adantl 388	. . . 4 |- ((A e. C /\ B e. D) -> (A ~< B -> (card` A) (_ (card` B)))
15		carden 4814	. . . . 5 |- ((A e. C /\ B e. D) -> ((card` A) = (card` B) <-> A ~~ B))
16		eqimss 2106	. . . . 5 |- ((card` A) = (card` B) -> (card` A) (_ (card` B))
17	15, 16	syl6bir 215	. . . 4 |- ((A e. C /\ B e. D) -> (A ~~ B -> (card` A) (_ (card` B)))
18	14, 17	jaod 424	. . 3 |- ((A e. C /\ B e. D) -> ((A ~< B \/ A ~~ B) -> (card` A) (_ (card` B)))
19		brdom2 4378	. . 3 |- (A ~<_ B <-> (A ~< B \/ A ~~ B))
20	18, 19	syl5ib 206	. 2 |- ((A e. C /\ B e. D) -> (A ~<_ B -> (card` A) (_ (card` B)))
21	2, 20	impbid 515	1 |- ((A e. C /\ B e. D) -> ((card` A) (_ (card` B) <-> A ~<_ B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 955   e. wcel 957   (_ wss 2044   class class class wbr 2615  Oncon0 2944  ` cfv 3178   ~~ cen 4357   ~<_ cdom 4358   ~< csdm 4359  cardccrd 4796

This theorem is referenced by:  cardsdom 4820  domtri 4821  carduni 4841  cardprc 4844  cardaleph 4868  alephval2 4885

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-dom 4361  df-sdom 4362  df-card 4799

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