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Theorem cardval2 9932
Description: An alternate version of the value of the cardinal number of a set. Compare cardval 10487. This theorem could be used to give a simpler definition of card in place of df-card 9880. It apparently does not occur in the literature. (Contributed by NM, 7-Nov-2003.)
Assertion
Ref Expression
cardval2 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) = {π‘₯ ∈ On ∣ π‘₯ β‰Ί 𝐴})
Distinct variable group:   π‘₯,𝐴

Proof of Theorem cardval2
StepHypRef Expression
1 cardon 9885 . . . . . 6 (cardβ€˜π΄) ∈ On
21oneli 6432 . . . . 5 (π‘₯ ∈ (cardβ€˜π΄) β†’ π‘₯ ∈ On)
32pm4.71ri 562 . . . 4 (π‘₯ ∈ (cardβ€˜π΄) ↔ (π‘₯ ∈ On ∧ π‘₯ ∈ (cardβ€˜π΄)))
4 cardsdomel 9915 . . . . . 6 ((π‘₯ ∈ On ∧ 𝐴 ∈ dom card) β†’ (π‘₯ β‰Ί 𝐴 ↔ π‘₯ ∈ (cardβ€˜π΄)))
54ancoms 460 . . . . 5 ((𝐴 ∈ dom card ∧ π‘₯ ∈ On) β†’ (π‘₯ β‰Ί 𝐴 ↔ π‘₯ ∈ (cardβ€˜π΄)))
65pm5.32da 580 . . . 4 (𝐴 ∈ dom card β†’ ((π‘₯ ∈ On ∧ π‘₯ β‰Ί 𝐴) ↔ (π‘₯ ∈ On ∧ π‘₯ ∈ (cardβ€˜π΄))))
73, 6bitr4id 290 . . 3 (𝐴 ∈ dom card β†’ (π‘₯ ∈ (cardβ€˜π΄) ↔ (π‘₯ ∈ On ∧ π‘₯ β‰Ί 𝐴)))
87abbi2dv 2868 . 2 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) = {π‘₯ ∣ (π‘₯ ∈ On ∧ π‘₯ β‰Ί 𝐴)})
9 df-rab 3407 . 2 {π‘₯ ∈ On ∣ π‘₯ β‰Ί 𝐴} = {π‘₯ ∣ (π‘₯ ∈ On ∧ π‘₯ β‰Ί 𝐴)}
108, 9eqtr4di 2791 1 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) = {π‘₯ ∈ On ∣ π‘₯ β‰Ί 𝐴})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  {crab 3406   class class class wbr 5106  dom cdm 5634  Oncon0 6318  β€˜cfv 6497   β‰Ί csdm 8885  cardccrd 9876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-card 9880
This theorem is referenced by:  ondomon  10504  alephsuc3  10521
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