ecdmn0 - Metamath Proof Explorer

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Theorem ecdmn0 8726

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 4319	. . 3 ⊢ ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)
3		ecexr 8679	. . . 4 ⊢ (𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V)
4	3	exlimiv 1930	. . 3 ⊢ (∃𝑥 𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V)
5	2, 4	sylbi 217	. 2 ⊢ ([𝐴]𝑅 ≠ ∅ → 𝐴 ∈ V)
6		vex 3454	. . . . 5 ⊢ 𝑥 ∈ V
7		elecg 8718	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
8	6, 7	mpan 690	. . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
9	8	exbidv 1921	. . 3 ⊢ (𝐴 ∈ V → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
10	2	a1i 11	. . 3 ⊢ (𝐴 ∈ V → ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅))
11		eldmg 5865	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
12	9, 10, 11	3bitr4rd 312	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅))
13	1, 5, 12	pm5.21nii 378	1 ⊢ (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∃wex 1779 ∈ wcel 2109 ≠ wne 2926 Vcvv 3450 ∅c0 4299 class class class wbr 5110 dom cdm 5641 [cec 8672

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2709 df-cleq 2722 df-clel 2804 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-sn 4593 df-pr 4595 df-op 4599 df-br 5111 df-opab 5173 df-xp 5647 df-cnv 5649 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-ec 8676

This theorem is referenced by: ereldm 8727 elqsn0 8760 ecelqsdm 8761 eceqoveq 8798 divsfval 17517 sylow1lem5 19539 vitalilem2 25517 vitalilem3 25518 dfdm6 38296 dmecd 38299 n0elqs 38321

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