Theorem carddomi 4807

Description: Two sets have the dominance relationship if their cardinalities have the subset relationship.

Assertion

Ref	Expression
carddomi	|- (A e. C -> ((card` A) (_ (card` B) -> A ~<_ B))

Proof of Theorem carddomi

Step	Hyp	Ref	Expression
1		endomtr 4401	. . . 4 |- ((A ~~ (card` A) /\ (card` A) ~<_ (card` B)) -> A ~<_ (card` B))
2		cardid 4800	. . . . 5 |- (card` A) ~~ A
3		ensymg 4392	. . . . 5 |- (A e. C -> ((card` A) ~~ A -> A ~~ (card` A)))
4	2, 3	mpi 44	. . . 4 |- (A e. C -> A ~~ (card` A))
5		cardon 4799	. . . . 5 |- (card` A) e. On
6		ssdomg 4389	. . . . 5 |- ((card` A) e. On -> ((card` A) (_ (card` B) -> (card` A) ~<_ (card` B)))
7	5, 6	ax-mp 7	. . . 4 |- ((card` A) (_ (card` B) -> (card` A) ~<_ (card` B))
8	1, 4, 7	syl2an 454	. . 3 |- ((A e. C /\ (card` A) (_ (card` B)) -> A ~<_ (card` B))
9		cardid 4800	. . . 4 |- (card` B) ~~ B
10		domentr 4402	. . . 4 |- ((A ~<_ (card` B) /\ (card` B) ~~ B) -> A ~<_ B)
11	9, 10	mpan2 694	. . 3 |- (A ~<_ (card` B) -> A ~<_ B)
12	8, 11	syl 10	. 2 |- ((A e. C /\ (card` A) (_ (card` B)) -> A ~<_ B)
13	12	ex 373	1 |- (A e. C -> ((card` A) (_ (card` B) -> A ~<_ B))

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 955   (_ wss 2037   class class class wbr 2609  Oncon0 2938  ` cfv 3172   ~~ cen 4348   ~<_ cdom 4349  cardccrd 4785

This theorem is referenced by:  carddom 4808

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-card 4788

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