Theorem carddom 4808

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 4807	. . 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 4807	. . . . . . . . 9 |- (B e. D -> ((card` B) (_ (card` A) -> B ~<_ A))
4		cardon 4799	. . . . . . . . . 10 |- (card` A) e. On
5	4	onelss 3090	. . . . . . . . 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 4443	. . . . . . . 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 4799	. . . . . . 7 |- (card` B) e. On
11		ontri1 2971	. . . . . . 7 |- (((card` A) e. On /\ (card` B) e. On) -> ((card` A) (_ (card` B) <-> -. (card` B) e. (card` A)))
12	4, 10, 11	mp2an 695	. . . . . 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 4803	. . . . 5 |- ((A e. C /\ B e. D) -> ((card` A) = (card` B) <-> A ~~ B))
16		eqimss 2099	. . . . 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 4369	. . 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 514	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 953   e. wcel 955   (_ wss 2037   class class class wbr 2609  Oncon0 2938  ` cfv 3172   ~~ cen 4348   ~<_ cdom 4349   ~< csdm 4350  cardccrd 4785

This theorem is referenced by:  cardsdom 4809  domtri 4810  carduni 4830  cardprc 4833  cardaleph 4857  alephval2 4874

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-ac 4716

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-er 4245  df-en 4351  df-dom 4352  df-sdom 4353  df-card 4788

Copyright terms: Public domain