ecdmn0 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ecdmn0		Structured version Visualization version GIF version

Theorem ecdmn0 8684

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 3459	. 2 ⊢ (𝐴 ∈ dom 𝑅 → 𝐴 ∈ V)
2		n0 4306	. . 3 ⊢ ([𝐴]𝑅 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)
3		ecexr 8637	. . . 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 3442	. . . . 5 ⊢ 𝑥 ∈ V
7		elecg 8676	. . . . 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 5845	. . 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 2925 Vcvv 3438 ∅c0 4286 class class class wbr 5095 dom cdm 5623 [cec 8630

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 2701 ax-sep 5238 ax-nul 5248 ax-pr 5374

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 2708 df-cleq 2721 df-clel 2803 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3397 df-v 3440 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-sn 4580 df-pr 4582 df-op 4586 df-br 5096 df-opab 5158 df-xp 5629 df-cnv 5631 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-ec 8634

This theorem is referenced by: ereldm 8685 elqsn0 8718 ecelqsdm 8719 eceqoveq 8756 divsfval 17469 sylow1lem5 19499 vitalilem2 25526 vitalilem3 25527 dfdm6 38274 dmecd 38277 n0elqs 38299

W3C validator