Theorem cardsdomel 4852

Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93 (use cardsdom 4837 to obtain the exact proposition from this one).

Assertion

Ref	Expression
cardsdomel	|- (A e. On -> (A ~< B <-> A e. (card` B)))

Proof of Theorem cardsdomel

Step	Hyp	Ref	Expression
1		ssdom2g 4409	. . . . 5 |- (A e. On -> ((card` B) (_ A -> (card` B) ~<_ A))
2		cardon 4827	. . . . . 6 |- (card` B) e. On
3		ontri1 2981	. . . . . 6 |- (((card` B) e. On /\ A e. On) -> ((card` B) (_ A <-> -. A e. (card` B)))
4	2, 3	mpan 695	. . . . 5 |- (A e. On -> ((card` B) (_ A <-> -. A e. (card` B)))
5		domtri 4838	. . . . . 6 |- (((card` B) e. On /\ A e. On) -> ((card` B) ~<_ A <-> -. A ~< (card` B)))
6	2, 5	mpan 695	. . . . 5 |- (A e. On -> ((card` B) ~<_ A <-> -. A ~< (card` B)))
7	1, 4, 6	3imtr3d 542	. . . 4 |- (A e. On -> (-. A e. (card` B) -> -. A ~< (card` B)))
8	7	a3d 75	. . 3 |- (A e. On -> (A ~< (card` B) -> A e. (card` B)))
9	2	onelss 3100	. . . . . 6 |- (A e. (card` B) -> A (_ (card` B))
10		ssdom2g 4409	. . . . . . 7 |- ((card` B) e. On -> (A (_ (card` B) -> A ~<_ (card` B)))
11	2, 10	ax-mp 7	. . . . . 6 |- (A (_ (card` B) -> A ~<_ (card` B))
12	9, 11	syl 10	. . . . 5 |- (A e. (card` B) -> A ~<_ (card` B))
13		cardidm 4849	. . . . . . 7 |- (card` (card` B)) = (card` B)
14	13	eleq2i 1538	. . . . . 6 |- (A e. (card` (card` B)) <-> A e. (card` B))
15		cardne 4830	. . . . . 6 |- (A e. (card` (card` B)) -> -. A ~~ (card` B))
16	14, 15	sylbir 201	. . . . 5 |- (A e. (card` B) -> -. A ~~ (card` B))
17	12, 16	jca 288	. . . 4 |- (A e. (card` B) -> (A ~<_ (card` B) /\ -. A ~~ (card` B)))
18		brsdom 4381	. . . 4 |- (A ~< (card` B) <-> (A ~<_ (card` B) /\ -. A ~~ (card` B)))
19	17, 18	sylibr 200	. . 3 |- (A e. (card` B) -> A ~< (card` B))
20	8, 19	impbid1 517	. 2 |- (A e. On -> (A ~< (card` B) <-> A e. (card` B)))
21		sdomsdomcard 4848	. 2 |- (A ~< B <-> A ~< (card` B))
22	20, 21	syl5bb 532	1 |- (A e. On -> (A ~< B <-> A e. (card` B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 958   (_ wss 2047   class class class wbr 2619  Oncon0 2948  ` cfv 3182   ~~ cen 4364   ~<_ cdom 4365   ~< csdm 4366  cardccrd 4813

This theorem is referenced by:  iscard 4853  cardval2 4855  alephnbtwn 4868  alephnbtwn2 4869  alephord2 4876  alephval2 4902

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-er 4261  df-en 4368  df-dom 4369  df-sdom 4370  df-card 4816

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