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Theorem ecdmn0 8696
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 3464 . 2 (𝐴 ∈ dom 𝑅𝐴 ∈ V)
2 n0 4307 . . 3 ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)
3 ecexr 8654 . . . 4 (𝑥 ∈ [𝐴]𝑅𝐴 ∈ V)
43exlimiv 1934 . . 3 (∃𝑥 𝑥 ∈ [𝐴]𝑅𝐴 ∈ V)
52, 4sylbi 216 . 2 ([𝐴]𝑅 ≠ ∅ → 𝐴 ∈ V)
6 vex 3450 . . . . 5 𝑥 ∈ V
7 elecg 8692 . . . . 5 ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
86, 7mpan 689 . . . 4 (𝐴 ∈ V → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
98exbidv 1925 . . 3 (𝐴 ∈ V → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
102a1i 11 . . 3 (𝐴 ∈ V → ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅))
11 eldmg 5855 . . 3 (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
129, 10, 113bitr4rd 312 . 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 205  wex 1782  wcel 2107  wne 2944  Vcvv 3446  c0 4283   class class class wbr 5106  dom cdm 5634  [cec 8647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ec 8651
This theorem is referenced by:  ereldm  8697  elqsn0  8726  ecelqsdm  8727  eceqoveq  8762  divsfval  17430  sylow1lem5  19385  vitalilem2  24976  vitalilem3  24977  dfdm6  36765  dmecd  36768  n0elqs  36790
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