Theorem cardsdomel 4832

Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93 (use cardsdom 4817 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 4396	. . . . 5 |- (A e. On -> ((card` B) (_ A -> (card` B) ~<_ A))
2		cardon 4807	. . . . . 6 |- (card` B) e. On
3		ontri1 2976	. . . . . 6 |- (((card` B) e. On /\ A e. On) -> ((card` B) (_ A <-> -. A e. (card` B)))
4	2, 3	mpan 694	. . . . 5 |- (A e. On -> ((card` B) (_ A <-> -. A e. (card` B)))
5		domtri 4818	. . . . . 6 |- (((card` B) e. On /\ A e. On) -> ((card` B) ~<_ A <-> -. A ~< (card` B)))
6	2, 5	mpan 694	. . . . 5 |- (A e. On -> ((card` B) ~<_ A <-> -. A ~< (card` B)))
7	1, 4, 6	3imtr3d 541	. . . 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 3095	. . . . . 6 |- (A e. (card` B) -> A (_ (card` B))
10		ssdom2g 4396	. . . . . . 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 4829	. . . . . . 7 |- (card` (card` B)) = (card` B)
14	13	eleq2i 1535	. . . . . 6 |- (A e. (card` (card` B)) <-> A e. (card` B))
15		cardne 4810	. . . . . 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 4369	. . . 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 516	. 2 |- (A e. On -> (A ~< (card` B) <-> A e. (card` B)))
21		sdomsdomcard 4828	. 2 |- (A ~< B <-> A ~< (card` B))
22	20, 21	syl5bb 531	1 |- (A e. On -> (A ~< B <-> A e. (card` B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 956   (_ wss 2043   class class class wbr 2614  Oncon0 2943  ` cfv 3177   ~~ cen 4354   ~<_ cdom 4355   ~< csdm 4356  cardccrd 4793

This theorem is referenced by:  iscard 4833  cardval2 4835  alephnbtwn 4848  alephnbtwn2 4849  alephord2 4856  alephval2 4882

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-ac 4724

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-suc 2949  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-er 4251  df-en 4357  df-dom 4358  df-sdom 4359  df-card 4796

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