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Mirrors > Home > MPE Home > Th. List > eq0 | Structured version Visualization version GIF version |
Description: A class is equal to the empty set if and only if it has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.) Avoid ax-11 2155, ax-12 2172. (Revised by Gino Giotto and Steven Nguyen, 28-Jun-2024.) Avoid ax-8 2109, df-clel 2811. (Revised by Gino Giotto, 6-Sep-2024.) |
Ref | Expression |
---|---|
eq0 | ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfnul4 4325 | . . 3 ⊢ ∅ = {𝑦 ∣ ⊥} | |
2 | 1 | eqeq2i 2746 | . 2 ⊢ (𝐴 = ∅ ↔ 𝐴 = {𝑦 ∣ ⊥}) |
3 | dfcleq 2726 | . . 3 ⊢ (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥})) | |
4 | df-clab 2711 | . . . . . . 7 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ [𝑥 / 𝑦]⊥) | |
5 | sbv 2092 | . . . . . . 7 ⊢ ([𝑥 / 𝑦]⊥ ↔ ⊥) | |
6 | 4, 5 | bitri 275 | . . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ ⊥) |
7 | 6 | bibi2i 338 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ (𝑥 ∈ 𝐴 ↔ ⊥)) |
8 | nbfal 1557 | . . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ ⊥)) | |
9 | 7, 8 | bitr4i 278 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ ¬ 𝑥 ∈ 𝐴) |
10 | 9 | albii 1822 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
11 | 3, 10 | bitri 275 | . 2 ⊢ (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
12 | 2, 11 | bitri 275 | 1 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∀wal 1540 = wceq 1542 ⊥wfal 1554 [wsb 2068 ∈ wcel 2107 {cab 2710 ∅c0 4323 |
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-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-dif 3952 df-nul 4324 |
This theorem is referenced by: neq0 4346 nel0 4351 0el 4361 ssdif0 4364 difin0ss 4369 inssdif0 4370 ralf0OLD 4518 disjiun 5136 0ex 5308 reldm0 5928 iresn0n0 6054 uzwo 12895 hashgt0elex 14361 hausdiag 23149 rnelfmlem 23456 prv0 34421 wzel 34796 knoppndv 35410 bj-nul 35937 bj-nuliota 35938 bj-nuliotaALT 35939 nninfnub 36619 prtlem14 37744 orddif0suc 42018 nrhmzr 46647 zrninitoringc 46969 |
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