Theorem ecdmn0 8773

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.

Step	Hyp	Ref	Expression
1		elex 3471	. 2 ⊢ (𝐴 ∈ dom 𝑅 → 𝐴 ∈ V)
2		n0 4299	. . 3 ⊢ ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)
3		ecexr 8725	. . . 4 ⊢ (𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V)
4	3	exlimiv 1962	. . 3 ⊢ (∃𝑥 𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V)
5	2, 4	sylbi 220	. 2 ⊢ ([𝐴]𝑅 ≠ ∅ → 𝐴 ∈ V)
6		vex 3454	. . . . 5 ⊢ 𝑥 ∈ V
7		elecg 8765	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
8	6, 7	mpan 703	. . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
9	8	exbidv 1953	. . 3 ⊢ (𝐴 ∈ V → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
10	2	a1i 11	. . 3 ⊢ (𝐴 ∈ V → ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅))
11		eldmg 5879	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
12	9, 10, 11	3bitr4rd 315	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅))
13	1, 5, 12	pm5.21nii 381	1 ⊢ (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅)

This proof depends on syntax axioms:   ↔ wb 209  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  Vcvv 3450  ∅c0 4278   class class class wbr 5102  dom cdm 5650  [cec 8718

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ec 8722

This theorem is used by:  ereldm  8774  elqsn0  8808  ecelqsdm  8809  eceqoveq  8846  divsfval  17724  sylow1lem5  19821  vitalilem2  25935  vitalilem3  25936  dfdm6  39332  dmecd  39335  n0elqs  39357  disjimeceqim  39829