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Mirrors > Home > MPE Home > Th. List > Mathboxes > n0elqs | Structured version Visualization version GIF version |
Description: Two ways of expressing that the empty set is not an element of a quotient set. (Contributed by Peter Mazsa, 5-Dec-2019.) |
Ref | Expression |
---|---|
n0elqs | ⊢ (¬ ∅ ∈ (𝐴 / 𝑅) ↔ 𝐴 ⊆ dom 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecdmn0 8325 | . . 3 ⊢ (𝑥 ∈ dom 𝑅 ↔ [𝑥]𝑅 ≠ ∅) | |
2 | 1 | ralbii 3162 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ dom 𝑅 ↔ ∀𝑥 ∈ 𝐴 [𝑥]𝑅 ≠ ∅) |
3 | dfss3 3953 | . 2 ⊢ (𝐴 ⊆ dom 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ dom 𝑅) | |
4 | nne 3017 | . . . . 5 ⊢ (¬ [𝑥]𝑅 ≠ ∅ ↔ [𝑥]𝑅 = ∅) | |
5 | 4 | rexbii 3244 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ [𝑥]𝑅 ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 [𝑥]𝑅 = ∅) |
6 | 5 | notbii 321 | . . 3 ⊢ (¬ ∃𝑥 ∈ 𝐴 ¬ [𝑥]𝑅 ≠ ∅ ↔ ¬ ∃𝑥 ∈ 𝐴 [𝑥]𝑅 = ∅) |
7 | dfral2 3234 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 [𝑥]𝑅 ≠ ∅ ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ [𝑥]𝑅 ≠ ∅) | |
8 | 0ex 5202 | . . . . . 6 ⊢ ∅ ∈ V | |
9 | 8 | elqs 8338 | . . . . 5 ⊢ (∅ ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 ∅ = [𝑥]𝑅) |
10 | eqcom 2825 | . . . . . 6 ⊢ (∅ = [𝑥]𝑅 ↔ [𝑥]𝑅 = ∅) | |
11 | 10 | rexbii 3244 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∅ = [𝑥]𝑅 ↔ ∃𝑥 ∈ 𝐴 [𝑥]𝑅 = ∅) |
12 | 9, 11 | bitri 276 | . . . 4 ⊢ (∅ ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 [𝑥]𝑅 = ∅) |
13 | 12 | notbii 321 | . . 3 ⊢ (¬ ∅ ∈ (𝐴 / 𝑅) ↔ ¬ ∃𝑥 ∈ 𝐴 [𝑥]𝑅 = ∅) |
14 | 6, 7, 13 | 3bitr4ri 305 | . 2 ⊢ (¬ ∅ ∈ (𝐴 / 𝑅) ↔ ∀𝑥 ∈ 𝐴 [𝑥]𝑅 ≠ ∅) |
15 | 2, 3, 14 | 3bitr4ri 305 | 1 ⊢ (¬ ∅ ∈ (𝐴 / 𝑅) ↔ 𝐴 ⊆ dom 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ∃wrex 3136 ⊆ wss 3933 ∅c0 4288 dom cdm 5548 [cec 8276 / cqs 8277 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ec 8280 df-qs 8284 |
This theorem is referenced by: n0elqs2 35465 n0eldmqs 35763 |
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