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Theorem ecdmn0 8793
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 3499 . 2 (𝐴 ∈ dom 𝑅𝐴 ∈ V)
2 n0 4359 . . 3 ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)
3 ecexr 8749 . . . 4 (𝑥 ∈ [𝐴]𝑅𝐴 ∈ V)
43exlimiv 1928 . . 3 (∃𝑥 𝑥 ∈ [𝐴]𝑅𝐴 ∈ V)
52, 4sylbi 217 . 2 ([𝐴]𝑅 ≠ ∅ → 𝐴 ∈ V)
6 vex 3482 . . . . 5 𝑥 ∈ V
7 elecg 8788 . . . . 5 ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
86, 7mpan 690 . . . 4 (𝐴 ∈ V → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
98exbidv 1919 . . 3 (𝐴 ∈ V → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
102a1i 11 . . 3 (𝐴 ∈ V → ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅))
11 eldmg 5912 . . 3 (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
129, 10, 113bitr4rd 312 . 2 (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅))
131, 5, 12pm5.21nii 378 1 (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wex 1776  wcel 2106  wne 2938  Vcvv 3478  c0 4339   class class class wbr 5148  dom cdm 5689  [cec 8742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746
This theorem is referenced by:  ereldm  8794  elqsn0  8825  ecelqsdm  8826  eceqoveq  8861  divsfval  17594  sylow1lem5  19635  vitalilem2  25658  vitalilem3  25659  dfdm6  38283  dmecd  38286  n0elqs  38308
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