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Theorem cardid2 9894
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 9893 . . 3 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴})
2 ssrab2 4038 . . . 4 {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} βŠ† On
3 fvex 6856 . . . . . 6 (cardβ€˜π΄) ∈ V
41, 3eqeltrrdi 2843 . . . . 5 (𝐴 ∈ dom card β†’ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ∈ V)
5 intex 5295 . . . . 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 7726 . . . 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 588 . . 3 (𝐴 ∈ dom card β†’ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴})
91, 8eqeltrd 2834 . 2 (𝐴 ∈ dom card β†’ (cardβ€˜π΄) ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴})
10 breq1 5109 . . . 4 (𝑦 = (cardβ€˜π΄) β†’ (𝑦 β‰ˆ 𝐴 ↔ (cardβ€˜π΄) β‰ˆ 𝐴))
1110elrab 3646 . . 3 ((cardβ€˜π΄) ∈ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴} ↔ ((cardβ€˜π΄) ∈ On ∧ (cardβ€˜π΄) β‰ˆ 𝐴))
1211simprbi 498 . 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 2107   β‰  wne 2940  {crab 3406  Vcvv 3444   βŠ† wss 3911  βˆ…c0 4283  βˆ© cint 4908   class class class wbr 5106  dom cdm 5634  Oncon0 6318  β€˜cfv 6497   β‰ˆ cen 8883  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-pr 5385
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-fv 6505  df-en 8887  df-card 9880
This theorem is referenced by:  isnum3  9895  oncardid  9897  cardidm  9900  ficardom  9902  ficardid  9903  cardnn  9904  cardnueq0  9905  carden2a  9907  carden2b  9908  carddomi2  9911  sdomsdomcardi  9912  cardsdomelir  9914  cardsdomel  9915  infxpidm2  9958  dfac8b  9972  numdom  9979  alephnbtwn2  10013  alephsucdom  10020  infenaleph  10032  dfac12r  10087  cardadju  10135  pwsdompw  10145  cff1  10199  cfflb  10200  cflim2  10204  cfss  10206  cfslb  10207  domtriomlem  10383  cardid  10488  cardidg  10489  carden  10492  sdomsdomcard  10501  hargch  10614  gch2  10616  hashkf  14238
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