MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cardval2 Structured version   Visualization version   GIF version

Theorem cardval2 9027
Description: An alternate version of the value of the cardinal number of a set. Compare cardval 9580. This theorem could be used to give us a simpler definition of card in place of df-card 8975. It apparently does not occur in the literature. (Contributed by NM, 7-Nov-2003.)
Assertion
Ref Expression
cardval2 (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
Distinct variable group:   𝑥,𝐴

Proof of Theorem cardval2
StepHypRef Expression
1 cardsdomel 9010 . . . . . 6 ((𝑥 ∈ On ∧ 𝐴 ∈ dom card) → (𝑥𝐴𝑥 ∈ (card‘𝐴)))
21ancoms 468 . . . . 5 ((𝐴 ∈ dom card ∧ 𝑥 ∈ On) → (𝑥𝐴𝑥 ∈ (card‘𝐴)))
32pm5.32da 676 . . . 4 (𝐴 ∈ dom card → ((𝑥 ∈ On ∧ 𝑥𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ (card‘𝐴))))
4 cardon 8980 . . . . . 6 (card‘𝐴) ∈ On
54oneli 5996 . . . . 5 (𝑥 ∈ (card‘𝐴) → 𝑥 ∈ On)
65pm4.71ri 668 . . . 4 (𝑥 ∈ (card‘𝐴) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ (card‘𝐴)))
73, 6syl6rbbr 279 . . 3 (𝐴 ∈ dom card → (𝑥 ∈ (card‘𝐴) ↔ (𝑥 ∈ On ∧ 𝑥𝐴)))
87abbi2dv 2880 . 2 (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥𝐴)})
9 df-rab 3059 . 2 {𝑥 ∈ On ∣ 𝑥𝐴} = {𝑥 ∣ (𝑥 ∈ On ∧ 𝑥𝐴)}
108, 9syl6eqr 2812 1 (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  {cab 2746  {crab 3054   class class class wbr 4804  dom cdm 5266  Oncon0 5884  cfv 6049  csdm 8122  cardccrd 8971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-card 8975
This theorem is referenced by:  ondomon  9597  alephsuc3  9614
  Copyright terms: Public domain W3C validator