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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 8191 | . . 3 ⊢ (𝑥 ∈ dom 𝑅 ↔ [𝑥]𝑅 ≠ ∅) | |
2 | 1 | ralbii 3132 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ dom 𝑅 ↔ ∀𝑥 ∈ 𝐴 [𝑥]𝑅 ≠ ∅) |
3 | dfss3 3882 | . 2 ⊢ (𝐴 ⊆ dom 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ dom 𝑅) | |
4 | nne 2988 | . . . . 5 ⊢ (¬ [𝑥]𝑅 ≠ ∅ ↔ [𝑥]𝑅 = ∅) | |
5 | 4 | rexbii 3211 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ [𝑥]𝑅 ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 [𝑥]𝑅 = ∅) |
6 | 5 | notbii 321 | . . 3 ⊢ (¬ ∃𝑥 ∈ 𝐴 ¬ [𝑥]𝑅 ≠ ∅ ↔ ¬ ∃𝑥 ∈ 𝐴 [𝑥]𝑅 = ∅) |
7 | dfral2 3201 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 [𝑥]𝑅 ≠ ∅ ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ [𝑥]𝑅 ≠ ∅) | |
8 | 0ex 5107 | . . . . . 6 ⊢ ∅ ∈ V | |
9 | 8 | elqs 8204 | . . . . 5 ⊢ (∅ ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 ∅ = [𝑥]𝑅) |
10 | eqcom 2802 | . . . . . 6 ⊢ (∅ = [𝑥]𝑅 ↔ [𝑥]𝑅 = ∅) | |
11 | 10 | rexbii 3211 | . . . . 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 1522 ∈ wcel 2081 ≠ wne 2984 ∀wral 3105 ∃wrex 3106 ⊆ wss 3863 ∅c0 4215 dom cdm 5448 [cec 8142 / cqs 8143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5099 ax-nul 5106 ax-pr 5226 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3710 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-nul 4216 df-if 4386 df-sn 4477 df-pr 4479 df-op 4483 df-br 4967 df-opab 5029 df-xp 5454 df-cnv 5456 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-ec 8146 df-qs 8150 |
This theorem is referenced by: n0elqs2 35142 n0eldmqs 35440 |
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