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Theorem ecdmn0 8329
Description: A representative of a nonempty equivalence class belongs to the domain of the equivalence relation. (Contributed by NM, 15-Feb-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
ecdmn0 (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅)

Proof of Theorem ecdmn0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elex 3517 . 2 (𝐴 ∈ dom 𝑅𝐴 ∈ V)
2 n0 4313 . . 3 ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)
3 ecexr 8287 . . . 4 (𝑥 ∈ [𝐴]𝑅𝐴 ∈ V)
43exlimiv 1924 . . 3 (∃𝑥 𝑥 ∈ [𝐴]𝑅𝐴 ∈ V)
52, 4sylbi 218 . 2 ([𝐴]𝑅 ≠ ∅ → 𝐴 ∈ V)
6 vex 3502 . . . . 5 𝑥 ∈ V
7 elecg 8325 . . . . 5 ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
86, 7mpan 686 . . . 4 (𝐴 ∈ V → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
98exbidv 1915 . . 3 (𝐴 ∈ V → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
102a1i 11 . . 3 (𝐴 ∈ V → ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅))
11 eldmg 5765 . . 3 (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
129, 10, 113bitr4rd 313 . 2 (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅))
131, 5, 12pm5.21nii 380 1 (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wex 1773  wcel 2107  wne 3020  Vcvv 3499  c0 4294   class class class wbr 5062  dom cdm 5553  [cec 8280
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-xp 5559  df-cnv 5561  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-ec 8284
This theorem is referenced by:  ereldm  8330  elqsn0  8359  ecelqsdm  8360  eceqoveq  8395  divsfval  16812  sylow1lem5  18649  vitalilem2  24125  vitalilem3  24126  dfdm6  35427  dmecd  35430  n0elqs  35451
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