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Theorem cardid2 9947
Description: Any numerable set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
Assertion
Ref Expression
cardid2 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) β‰ˆ 𝐴)

Proof of Theorem cardid2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cardval3 9946 . . 3 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴})
2 ssrab2 4072 . . . 4 {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} βŠ† On
3 fvex 6897 . . . . . 6 (cardβ€˜π΄) ∈ V
41, 3eqeltrrdi 2836 . . . . 5 (𝐴 ∈ dom card β†’ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ∈ V)
5 intex 5330 . . . . 5 ({𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} β‰  βˆ… ↔ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ∈ V)
64, 5sylibr 233 . . . 4 (𝐴 ∈ dom card β†’ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} β‰  βˆ…)
7 onint 7774 . . . 4 (({𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} βŠ† On ∧ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} β‰  βˆ…) β†’ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴})
82, 6, 7sylancr 586 . . 3 (𝐴 ∈ dom card β†’ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴})
91, 8eqeltrd 2827 . 2 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴})
10 breq1 5144 . . . 4 (𝑦 = (cardβ€˜π΄) β†’ (𝑦 β‰ˆ 𝐴 ↔ (cardβ€˜π΄) β‰ˆ 𝐴))
1110elrab 3678 . . 3 ((cardβ€˜π΄) ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ↔ ((cardβ€˜π΄) ∈ On ∧ (cardβ€˜π΄) β‰ˆ 𝐴))
1211simprbi 496 . 2 ((cardβ€˜π΄) ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
139, 12syl 17 1 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2098   β‰  wne 2934  {crab 3426  Vcvv 3468   βŠ† wss 3943  βˆ…c0 4317  βˆ© cint 4943   class class class wbr 5141  dom cdm 5669  Oncon0 6357  β€˜cfv 6536   β‰ˆ cen 8935  cardccrd 9929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-en 8939  df-card 9933
This theorem is referenced by:  isnum3  9948  oncardid  9950  cardidm  9953  ficardom  9955  ficardid  9956  cardnn  9957  cardnueq0  9958  carden2a  9960  carden2b  9961  carddomi2  9964  sdomsdomcardi  9965  cardsdomelir  9967  cardsdomel  9968  infxpidm2  10011  dfac8b  10025  numdom  10032  alephnbtwn2  10066  alephsucdom  10073  infenaleph  10085  dfac12r  10140  cardadju  10188  pwsdompw  10198  cff1  10252  cfflb  10253  cflim2  10257  cfss  10259  cfslb  10260  domtriomlem  10436  cardid  10541  cardidg  10542  carden  10545  sdomsdomcard  10554  hargch  10667  gch2  10669  hashkf  14294
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