ecdmn0 - Metamath Proof Explorer

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Theorem ecdmn0 8503

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 3440	. 2 ⊢ (𝐴 ∈ dom 𝑅 → 𝐴 ∈ V)
2		n0 4277	. . 3 ⊢ ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)
3		ecexr 8461	. . . 4 ⊢ (𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V)
4	3	exlimiv 1934	. . 3 ⊢ (∃𝑥 𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V)
5	2, 4	sylbi 216	. 2 ⊢ ([𝐴]𝑅 ≠ ∅ → 𝐴 ∈ V)
6		vex 3426	. . . . 5 ⊢ 𝑥 ∈ V
7		elecg 8499	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
8	6, 7	mpan 686	. . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
9	8	exbidv 1925	. . 3 ⊢ (𝐴 ∈ V → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
10	2	a1i 11	. . 3 ⊢ (𝐴 ∈ V → ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅))
11		eldmg 5796	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
12	9, 10, 11	3bitr4rd 311	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅))
13	1, 5, 12	pm5.21nii 379	1 ⊢ (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∃wex 1783 ∈ wcel 2108 ≠ wne 2942 Vcvv 3422 ∅c0 4253 class class class wbr 5070 dom cdm 5580 [cec 8454

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2817 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-br 5071 df-opab 5133 df-xp 5586 df-cnv 5588 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-ec 8458

This theorem is referenced by: ereldm 8504 elqsn0 8533 ecelqsdm 8534 eceqoveq 8569 divsfval 17175 sylow1lem5 19122 vitalilem2 24678 vitalilem3 24679 dfdm6 36364 dmecd 36367 n0elqs 36388

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