Theorem cardval2 4835

Description: An alternate version of the value of the cardinal number of a set. Compare cardval 4806. This theorem could be used to give us a simpler definition of card in place of df-card 4796. It apparently does not occur in the literature.

Assertion

Ref	Expression
cardval2	|- (card` A) = {x e. On | x ~< A}

Distinct variable group:   x,A

Proof of Theorem cardval2

Step	Hyp	Ref	Expression
1		df-rab 1649	. 2 |- {x e. On | x ~< A} = {x | (x e. On /\ x ~< A)}
2		cardon 4807	. . . . . 6 |- (card` A) e. On
3	2	onel 3093	. . . . 5 |- (x e. (card` A) -> x e. On)
4	3	pm4.71ri 637	. . . 4 |- (x e. (card` A) <-> (x e. On /\ x e. (card` A)))
5		cardsdomel 4832	. . . . 5 |- (x e. On -> (x ~< A <-> x e. (card` A)))
6	5	pm5.32i 644	. . . 4 |- ((x e. On /\ x ~< A) <-> (x e. On /\ x e. (card` A)))
7	4, 6	bitr4 176	. . 3 |- (x e. (card` A) <-> (x e. On /\ x ~< A))
8	7	abbii 1572	. 2 |- {x | x e. (card` A)} = {x | (x e. On /\ x ~< A)}
9		abid2 1577	. 2 |- {x | x e. (card` A)} = (card` A)
10	1, 8, 9	3eqtr2r 1499	1 |- (card` A) = {x e. On | x ~< A}

Syntax hints:   /\ wa 223   = wceq 954   e. wcel 956  {cab 1461  {crab 1645   class class class wbr 2614  Oncon0 2943  ` cfv 3177   ~< csdm 4356  cardccrd 4793

This theorem is referenced by:  ondomon 4836  alephsuc2 4861  alephsuc3 7535

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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