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Mirrors > Home > MPE Home > Th. List > eq0rdv | Structured version Visualization version GIF version |
Description: Deduction for equality to the empty set. (Contributed by NM, 11-Jul-2014.) Avoid ax-8 2109, df-clel 2811. (Revised by Gino Giotto, 6-Sep-2024.) |
Ref | Expression |
---|---|
eq0rdv.1 | ⊢ (𝜑 → ¬ 𝑥 ∈ 𝐴) |
Ref | Expression |
---|---|
eq0rdv | ⊢ (𝜑 → 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eq0rdv.1 | . . 3 ⊢ (𝜑 → ¬ 𝑥 ∈ 𝐴) | |
2 | 1 | alrimiv 1931 | . 2 ⊢ (𝜑 → ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
3 | dfnul4 4322 | . . . 4 ⊢ ∅ = {𝑦 ∣ ⊥} | |
4 | 3 | eqeq2i 2746 | . . 3 ⊢ (𝐴 = ∅ ↔ 𝐴 = {𝑦 ∣ ⊥}) |
5 | dfcleq 2726 | . . 3 ⊢ (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥})) | |
6 | df-clab 2711 | . . . . . . 7 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ [𝑥 / 𝑦]⊥) | |
7 | sbv 2092 | . . . . . . 7 ⊢ ([𝑥 / 𝑦]⊥ ↔ ⊥) | |
8 | 6, 7 | bitri 275 | . . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ ⊥) |
9 | 8 | bibi2i 338 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ (𝑥 ∈ 𝐴 ↔ ⊥)) |
10 | 9 | albii 1822 | . . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ ⊥)) |
11 | nbfal 1557 | . . . . . 6 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ ⊥)) | |
12 | 11 | bicomi 223 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ ⊥) ↔ ¬ 𝑥 ∈ 𝐴) |
13 | 12 | albii 1822 | . . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ ⊥) ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
14 | 10, 13 | bitri 275 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
15 | 4, 5, 14 | 3bitrri 298 | . 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ 𝐴 = ∅) |
16 | 2, 15 | sylib 217 | 1 ⊢ (𝜑 → 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∀wal 1540 = wceq 1542 ⊥wfal 1554 [wsb 2068 ∈ wcel 2107 {cab 2710 ∅c0 4320 |
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 3949 df-nul 4321 |
This theorem is referenced by: map0b 8865 disjen 9122 mapdom1 9130 pwxpndom2 10647 fzdisj 13515 smu01lem 16413 prmreclem5 16840 vdwap0 16896 natfval 17884 fucbas 17899 fuchom 17900 fuchomOLD 17901 coafval 18001 efgval 19569 lsppratlem6 20742 lbsextlem4 20751 psrvscafval 21480 cfinufil 23401 ufinffr 23402 fin1aufil 23405 bldisj 23873 reconnlem1 24311 pcofval 24495 bcthlem5 24814 volfiniun 25033 fta1g 25654 fta1 25790 rpvmasum 26996 0ringprmidl 32519 0ringmon1p 32581 0ringirng 32691 unblimceq0 35288 bj-ab0 35693 bj-projval 35782 finxpnom 36187 ipo0 43079 ifr0 43080 limclner 44240 |
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