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Theorem ecdmn0 8071
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 3413 . 2 (𝐴 ∈ dom 𝑅𝐴 ∈ V)
2 n0 4158 . . 3 ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)
3 ecexr 8031 . . . 4 (𝑥 ∈ [𝐴]𝑅𝐴 ∈ V)
43exlimiv 1973 . . 3 (∃𝑥 𝑥 ∈ [𝐴]𝑅𝐴 ∈ V)
52, 4sylbi 209 . 2 ([𝐴]𝑅 ≠ ∅ → 𝐴 ∈ V)
6 vex 3400 . . . . 5 𝑥 ∈ V
7 elecg 8067 . . . . 5 ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
86, 7mpan 680 . . . 4 (𝐴 ∈ V → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
98exbidv 1964 . . 3 (𝐴 ∈ V → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
102a1i 11 . . 3 (𝐴 ∈ V → ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅))
11 eldmg 5564 . . 3 (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
129, 10, 113bitr4rd 304 . 2 (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅))
131, 5, 12pm5.21nii 370 1 (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wex 1823  wcel 2106  wne 2968  Vcvv 3397  c0 4140   class class class wbr 4886  dom cdm 5355  [cec 8024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4887  df-opab 4949  df-xp 5361  df-cnv 5363  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-ec 8028
This theorem is referenced by:  ereldm  8072  elqsn0  8099  ecelqsdm  8100  eceqoveq  8136  divsfval  16593  sylow1lem5  18401  vitalilem2  23813  vitalilem3  23814  dfdm6  34685  dmecd  34688  n0elqs  34710
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