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Theorem carden2 10018
Description: Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 10582, the Axiom of Choice is not required. (Contributed by Mario Carneiro, 22-Sep-2013.)
Assertion
Ref Expression
carden2 ((𝐴 ∈ dom card ∧ 𝐡 ∈ dom card) β†’ ((cardβ€˜π΄) = (cardβ€˜π΅) ↔ 𝐴 β‰ˆ 𝐡))

Proof of Theorem carden2
StepHypRef Expression
1 carddom2 10008 . . 3 ((𝐴 ∈ dom card ∧ 𝐡 ∈ dom card) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ↔ 𝐴 β‰Ό 𝐡))
2 carddom2 10008 . . . 4 ((𝐡 ∈ dom card ∧ 𝐴 ∈ dom card) β†’ ((cardβ€˜π΅) βŠ† (cardβ€˜π΄) ↔ 𝐡 β‰Ό 𝐴))
32ancoms 457 . . 3 ((𝐴 ∈ dom card ∧ 𝐡 ∈ dom card) β†’ ((cardβ€˜π΅) βŠ† (cardβ€˜π΄) ↔ 𝐡 β‰Ό 𝐴))
41, 3anbi12d 630 . 2 ((𝐴 ∈ dom card ∧ 𝐡 ∈ dom card) β†’ (((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ∧ (cardβ€˜π΅) βŠ† (cardβ€˜π΄)) ↔ (𝐴 β‰Ό 𝐡 ∧ 𝐡 β‰Ό 𝐴)))
5 eqss 3997 . . 3 ((cardβ€˜π΄) = (cardβ€˜π΅) ↔ ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ∧ (cardβ€˜π΅) βŠ† (cardβ€˜π΄)))
65bicomi 223 . 2 (((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ∧ (cardβ€˜π΅) βŠ† (cardβ€˜π΄)) ↔ (cardβ€˜π΄) = (cardβ€˜π΅))
7 sbthb 9125 . 2 ((𝐴 β‰Ό 𝐡 ∧ 𝐡 β‰Ό 𝐴) ↔ 𝐴 β‰ˆ 𝐡)
84, 6, 73bitr3g 312 1 ((𝐴 ∈ dom card ∧ 𝐡 ∈ dom card) β†’ ((cardβ€˜π΄) = (cardβ€˜π΅) ↔ 𝐴 β‰ˆ 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   βŠ† wss 3949   class class class wbr 5152  dom cdm 5682  β€˜cfv 6553   β‰ˆ cen 8967   β‰Ό cdom 8968  cardccrd 9966
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-card 9970
This theorem is referenced by:  cardsdom2  10019  pm54.43lem  10031  sdom2en01  10333  fin23lem22  10358  fin1a2lem9  10439  pwfseqlem4  10693  hashen  14346
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