Theorem carddom 5987

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) C_ (card` B) <-> A ~<_ B))

Proof of Theorem carddom

Step	Hyp	Ref	Expression
1		carddomi 5986	. . 3 |- (A e. C -> ((card` A) C_ (card` B) -> A ~<_ B))
2	1	adantr 425	. 2 |- ((A e. C /\ B e. D) -> ((card` A) C_ (card` B) -> A ~<_ B))
3		carddomi 5986	. . . . . . . . 9 |- (B e. D -> ((card` B) C_ (card` A) -> B ~<_ A))
4		cardon 5976	. . . . . . . . . 10 |- (card` A) e. On
5	4	onelssi 3778	. . . . . . . . 9 |- ((card` B) e. (card` A) -> (card` B) C_ (card` A))
6	3, 5	syl5 20	. . . . . . . 8 |- (B e. D -> ((card` B) e. (card` A) -> B ~<_ A))
7		domnsym 5526	. . . . . . . 8 |- (B ~<_ A -> -. A ~< B)
8	6, 7	syl6 25	. . . . . . 7 |- (B e. D -> ((card` B) e. (card` A) -> -. A ~< B))
9	8	con2d 107	. . . . . 6 |- (B e. D -> (A ~< B -> -. (card` B) e. (card` A)))
10		cardon 5976	. . . . . . 7 |- (card` B) e. On
11		ontri1 3695	. . . . . . 7 |- (((card` A) e. On /\ (card` B) e. On) -> ((card` A) C_ (card` B) <-> -. (card` B) e. (card` A)))
12	4, 10, 11	mp2an 761	. . . . . 6 |- ((card` A) C_ (card` B) <-> -. (card` B) e. (card` A))
13	9, 12	syl6ibr 230	. . . . 5 |- (B e. D -> (A ~< B -> (card` A) C_ (card` B)))
14	13	adantl 424	. . . 4 |- ((A e. C /\ B e. D) -> (A ~< B -> (card` A) C_ (card` B)))
15		carden 5981	. . . . 5 |- ((A e. C /\ B e. D) -> ((card` A) = (card` B) <-> A ~~ B))
16		eqimss 2665	. . . . 5 |- ((card` A) = (card` B) -> (card` A) C_ (card` B))
17	15, 16	syl6bir 232	. . . 4 |- ((A e. C /\ B e. D) -> (A ~~ B -> (card` A) C_ (card` B)))
18	14, 17	jaod 469	. . 3 |- ((A e. C /\ B e. D) -> ((A ~< B \/ A ~~ B) -> (card` A) C_ (card` B)))
19		brdom2 5447	. . 3 |- (A ~<_ B <-> (A ~< B \/ A ~~ B))
20	18, 19	syl5ib 223	. 2 |- ((A e. C /\ B e. D) -> (A ~<_ B -> (card` A) C_ (card` B)))
21	2, 20	impbid 574	1 |- ((A e. C /\ B e. D) -> ((card` A) C_ (card` B) <-> A ~<_ B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300   C_ wss 2593   class class class wbr 3338  Oncon0 3657  ` cfv 3998   ~~ cen 5423   ~<_ cdom 5424   ~< csdm 5425  cardccrd 5859

This theorem is referenced by:  cardsdom 5988  domtri 5989  carduni 6010  cardprc 6013  cardaleph 6033  alephval2 6050

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-dom 5428  df-sdom 5429  df-card 5862

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