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Theorem ecdmn0 8380
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 3418 . 2 (𝐴 ∈ dom 𝑅𝐴 ∈ V)
2 n0 4245 . . 3 ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)
3 ecexr 8338 . . . 4 (𝑥 ∈ [𝐴]𝑅𝐴 ∈ V)
43exlimiv 1937 . . 3 (∃𝑥 𝑥 ∈ [𝐴]𝑅𝐴 ∈ V)
52, 4sylbi 220 . 2 ([𝐴]𝑅 ≠ ∅ → 𝐴 ∈ V)
6 vex 3404 . . . . 5 𝑥 ∈ V
7 elecg 8376 . . . . 5 ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
86, 7mpan 690 . . . 4 (𝐴 ∈ V → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
98exbidv 1928 . . 3 (𝐴 ∈ V → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
102a1i 11 . . 3 (𝐴 ∈ V → ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅))
11 eldmg 5751 . . 3 (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
129, 10, 113bitr4rd 315 . 2 (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅))
131, 5, 12pm5.21nii 383 1 (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wex 1786  wcel 2114  wne 2935  Vcvv 3400  c0 4221   class class class wbr 5040  dom cdm 5535  [cec 8331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-br 5041  df-opab 5103  df-xp 5541  df-cnv 5543  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-ec 8335
This theorem is referenced by:  ereldm  8381  elqsn0  8410  ecelqsdm  8411  eceqoveq  8446  divsfval  16936  sylow1lem5  18858  vitalilem2  24374  vitalilem3  24375  dfdm6  36093  dmecd  36096  n0elqs  36117
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