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Theorem carden2 9981
Description: Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 10545, 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 9971 . . 3 ((𝐴 ∈ dom card ∧ 𝐡 ∈ dom card) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ↔ 𝐴 β‰Ό 𝐡))
2 carddom2 9971 . . . 4 ((𝐡 ∈ dom card ∧ 𝐴 ∈ dom card) β†’ ((cardβ€˜π΅) βŠ† (cardβ€˜π΄) ↔ 𝐡 β‰Ό 𝐴))
32ancoms 458 . . 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 3992 . . 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 9093 . 2 ((𝐴 β‰Ό 𝐡 ∧ 𝐡 β‰Ό 𝐴) ↔ 𝐴 β‰ˆ 𝐡)
84, 6, 73bitr3g 313 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 395   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943   class class class wbr 5141  dom cdm 5669  β€˜cfv 6536   β‰ˆ cen 8935   β‰Ό cdom 8936  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-pow 5356  ax-pr 5420  ax-un 7721
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-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-card 9933
This theorem is referenced by:  cardsdom2  9982  pm54.43lem  9994  sdom2en01  10296  fin23lem22  10321  fin1a2lem9  10402  pwfseqlem4  10656  hashen  14309
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