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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 2108, df-clel 2810. (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 1930 | . 2 ⊢ (𝜑 → ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
3 | dfnul4 4324 | . . . 4 ⊢ ∅ = {𝑦 ∣ ⊥} | |
4 | 3 | eqeq2i 2745 | . . 3 ⊢ (𝐴 = ∅ ↔ 𝐴 = {𝑦 ∣ ⊥}) |
5 | dfcleq 2725 | . . 3 ⊢ (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥})) | |
6 | df-clab 2710 | . . . . . . 7 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ [𝑥 / 𝑦]⊥) | |
7 | sbv 2091 | . . . . . . 7 ⊢ ([𝑥 / 𝑦]⊥ ↔ ⊥) | |
8 | 6, 7 | bitri 274 | . . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ ⊥) |
9 | 8 | bibi2i 337 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ (𝑥 ∈ 𝐴 ↔ ⊥)) |
10 | 9 | albii 1821 | . . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ ⊥)) |
11 | nbfal 1556 | . . . . . 6 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ ⊥)) | |
12 | 11 | bicomi 223 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ ⊥) ↔ ¬ 𝑥 ∈ 𝐴) |
13 | 12 | albii 1821 | . . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ ⊥) ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
14 | 10, 13 | bitri 274 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
15 | 4, 5, 14 | 3bitrri 297 | . 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 1539 = wceq 1541 ⊥wfal 1553 [wsb 2067 ∈ wcel 2106 {cab 2709 ∅c0 4322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-dif 3951 df-nul 4323 |
This theorem is referenced by: map0b 8879 disjen 9136 mapdom1 9144 pwxpndom2 10662 fzdisj 13532 smu01lem 16430 prmreclem5 16857 vdwap0 16913 natfval 17901 fucbas 17916 fuchom 17917 fuchomOLD 17918 coafval 18018 efgval 19626 lsppratlem6 20910 lbsextlem4 20919 psrvscafval 21728 cfinufil 23652 ufinffr 23653 fin1aufil 23656 bldisj 24124 reconnlem1 24562 pcofval 24750 bcthlem5 25069 volfiniun 25288 fta1g 25909 fta1 26045 rpvmasum 27253 0ringprmidl 32830 0ringmon1p 32898 0ringirng 33030 unblimceq0 35686 bj-ab0 36091 bj-projval 36180 finxpnom 36585 ipo0 43510 ifr0 43511 limclner 44666 |
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