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Theorem cardnueq0 8775
Description: The empty set is the only numerable set with cardinality zero. (Contributed by Mario Carneiro, 7-Jan-2013.)
Assertion
Ref Expression
cardnueq0 (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))

Proof of Theorem cardnueq0
StepHypRef Expression
1 cardid2 8764 . . . 4 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
21ensymd 7992 . . 3 (𝐴 ∈ dom card → 𝐴 ≈ (card‘𝐴))
3 breq2 4648 . . . 4 ((card‘𝐴) = ∅ → (𝐴 ≈ (card‘𝐴) ↔ 𝐴 ≈ ∅))
4 en0 8004 . . . 4 (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
53, 4syl6bb 276 . . 3 ((card‘𝐴) = ∅ → (𝐴 ≈ (card‘𝐴) ↔ 𝐴 = ∅))
62, 5syl5ibcom 235 . 2 (𝐴 ∈ dom card → ((card‘𝐴) = ∅ → 𝐴 = ∅))
7 fveq2 6178 . . 3 (𝐴 = ∅ → (card‘𝐴) = (card‘∅))
8 card0 8769 . . 3 (card‘∅) = ∅
97, 8syl6eq 2670 . 2 (𝐴 = ∅ → (card‘𝐴) = ∅)
106, 9impbid1 215 1 (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1481  wcel 1988  c0 3907   class class class wbr 4644  dom cdm 5104  cfv 5876  cen 7937  cardccrd 8746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-er 7727  df-en 7941  df-card 8750
This theorem is referenced by:  carddomi2  8781  cfeq0  9063  cfsuc  9064  sdom2en01  9109
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