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Mathbox for Peter Mazsa |
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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 8750 | . . 3 ⊢ (𝑥 ∈ dom 𝑅 ↔ [𝑥]𝑅 ≠ ∅) | |
2 | 1 | ralbii 3094 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ dom 𝑅 ↔ ∀𝑥 ∈ 𝐴 [𝑥]𝑅 ≠ ∅) |
3 | dfss3 3971 | . 2 ⊢ (𝐴 ⊆ dom 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ dom 𝑅) | |
4 | nne 2945 | . . . . 5 ⊢ (¬ [𝑥]𝑅 ≠ ∅ ↔ [𝑥]𝑅 = ∅) | |
5 | 4 | rexbii 3095 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ [𝑥]𝑅 ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 [𝑥]𝑅 = ∅) |
6 | 5 | notbii 320 | . . 3 ⊢ (¬ ∃𝑥 ∈ 𝐴 ¬ [𝑥]𝑅 ≠ ∅ ↔ ¬ ∃𝑥 ∈ 𝐴 [𝑥]𝑅 = ∅) |
7 | dfral2 3100 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 [𝑥]𝑅 ≠ ∅ ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ [𝑥]𝑅 ≠ ∅) | |
8 | 0ex 5308 | . . . . . 6 ⊢ ∅ ∈ V | |
9 | 8 | elqs 8763 | . . . . 5 ⊢ (∅ ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 ∅ = [𝑥]𝑅) |
10 | eqcom 2740 | . . . . . 6 ⊢ (∅ = [𝑥]𝑅 ↔ [𝑥]𝑅 = ∅) | |
11 | 10 | rexbii 3095 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∅ = [𝑥]𝑅 ↔ ∃𝑥 ∈ 𝐴 [𝑥]𝑅 = ∅) |
12 | 9, 11 | bitri 275 | . . . 4 ⊢ (∅ ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 [𝑥]𝑅 = ∅) |
13 | 12 | notbii 320 | . . 3 ⊢ (¬ ∅ ∈ (𝐴 / 𝑅) ↔ ¬ ∃𝑥 ∈ 𝐴 [𝑥]𝑅 = ∅) |
14 | 6, 7, 13 | 3bitr4ri 304 | . 2 ⊢ (¬ ∅ ∈ (𝐴 / 𝑅) ↔ ∀𝑥 ∈ 𝐴 [𝑥]𝑅 ≠ ∅) |
15 | 2, 3, 14 | 3bitr4ri 304 | 1 ⊢ (¬ ∅ ∈ (𝐴 / 𝑅) ↔ 𝐴 ⊆ dom 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 ⊆ wss 3949 ∅c0 4323 dom cdm 5677 [cec 8701 / cqs 8702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-xp 5683 df-cnv 5685 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ec 8705 df-qs 8709 |
This theorem is referenced by: n0elqs2 37196 n0eldmqs 37518 |
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