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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 2814. (Revised by GG, 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 1925 | . 2 ⊢ (𝜑 → ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
3 | dfnul4 4341 | . . . 4 ⊢ ∅ = {𝑦 ∣ ⊥} | |
4 | 3 | eqeq2i 2748 | . . 3 ⊢ (𝐴 = ∅ ↔ 𝐴 = {𝑦 ∣ ⊥}) |
5 | dfcleq 2728 | . . 3 ⊢ (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥})) | |
6 | df-clab 2713 | . . . . . . 7 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ [𝑥 / 𝑦]⊥) | |
7 | sbv 2086 | . . . . . . 7 ⊢ ([𝑥 / 𝑦]⊥ ↔ ⊥) | |
8 | 6, 7 | bitri 275 | . . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ ⊥) |
9 | 8 | bibi2i 337 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ (𝑥 ∈ 𝐴 ↔ ⊥)) |
10 | 9 | albii 1816 | . . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ ⊥)) |
11 | nbfal 1552 | . . . . . 6 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ ⊥)) | |
12 | 11 | bicomi 224 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ ⊥) ↔ ¬ 𝑥 ∈ 𝐴) |
13 | 12 | albii 1816 | . . . 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 218 | 1 ⊢ (𝜑 → 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∀wal 1535 = wceq 1537 ⊥wfal 1549 [wsb 2062 ∈ wcel 2106 {cab 2712 ∅c0 4339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-dif 3966 df-nul 4340 |
This theorem is referenced by: map0b 8922 disjen 9173 mapdom1 9181 pwxpndom2 10703 fzdisj 13588 smu01lem 16519 prmreclem5 16954 vdwap0 17010 natfval 18001 fucbas 18016 fuchom 18017 fuchomOLD 18018 coafval 18118 efgval 19750 lsppratlem6 21172 lbsextlem4 21181 psrvscafval 21986 cfinufil 23952 ufinffr 23953 fin1aufil 23956 bldisj 24424 reconnlem1 24862 pcofval 25057 bcthlem5 25376 volfiniun 25596 fta1g 26224 fta1 26365 rpvmasum 27585 0ringprmidl 33457 0ringmon1p 33563 0ringirng 33704 unblimceq0 36490 bj-ab0 36891 bj-projval 36979 finxpnom 37384 ipo0 44445 ifr0 44446 limclner 45607 |
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