ecdmn0 - Metamath Proof Explorer

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Theorem ecdmn0 8812

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 3509	. 2 ⊢ (𝐴 ∈ dom 𝑅 → 𝐴 ∈ V)
2		n0 4376	. . 3 ⊢ ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)
3		ecexr 8768	. . . 4 ⊢ (𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V)
4	3	exlimiv 1929	. . 3 ⊢ (∃𝑥 𝑥 ∈ [𝐴]𝑅 → 𝐴 ∈ V)
5	2, 4	sylbi 217	. 2 ⊢ ([𝐴]𝑅 ≠ ∅ → 𝐴 ∈ V)
6		vex 3492	. . . . 5 ⊢ 𝑥 ∈ V
7		elecg 8807	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
8	6, 7	mpan 689	. . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝑥))
9	8	exbidv 1920	. . 3 ⊢ (𝐴 ∈ V → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
10	2	a1i 11	. . 3 ⊢ (𝐴 ∈ V → ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅))
11		eldmg 5923	. . 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 1777 ∈ wcel 2108 ≠ wne 2946 Vcvv 3488 ∅c0 4352 class class class wbr 5166 dom cdm 5700 [cec 8761

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-xp 5706 df-cnv 5708 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-ec 8765

This theorem is referenced by: ereldm 8813 elqsn0 8844 ecelqsdm 8845 eceqoveq 8880 divsfval 17607 sylow1lem5 19644 vitalilem2 25663 vitalilem3 25664 dfdm6 38257 dmecd 38260 n0elqs 38282

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