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Theorem isnumi 8732
Description: A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
isnumi ((𝐴 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ dom card)

Proof of Theorem isnumi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 breq1 4626 . . 3 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
21rspcev 3299 . 2 ((𝐴 ∈ On ∧ 𝐴𝐵) → ∃𝑥 ∈ On 𝑥𝐵)
3 isnum2 8731 . 2 (𝐵 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐵)
42, 3sylibr 224 1 ((𝐴 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wrex 2909   class class class wbr 4623  dom cdm 5084  Oncon0 5692  cen 7912  cardccrd 8721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-ord 5695  df-on 5696  df-fun 5859  df-fn 5860  df-f 5861  df-en 7916  df-card 8725
This theorem is referenced by:  finnum  8734  onenon  8735  tskwe  8736  xpnum  8737  isnum3  8740  dfac8alem  8812  cdanum  8981  fin67  9177  isfin7-2  9178  gch2  9457  gchacg  9462  znnen  14885  qnnen  14886  met1stc  22266  re2ndc  22544  uniiccdif  23286  dyadmbl  23308  opnmblALT  23311  mbfimaopnlem  23362  aannenlem3  24023  poimirlem32  33112
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