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Mirrors > Home > MPE Home > Th. List > eq0rdv | Structured version Visualization version GIF version |
Description: Deduction rule for equality to the empty set. (Contributed by NM, 11-Jul-2014.) |
Ref | Expression |
---|---|
eq0rdv.1 | ⊢ (𝜑 → ¬ 𝑥 ∈ 𝐴) |
Ref | Expression |
---|---|
eq0rdv | ⊢ (𝜑 → 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eq0rdv.1 | . . . 4 ⊢ (𝜑 → ¬ 𝑥 ∈ 𝐴) | |
2 | 1 | pm2.21d 117 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∅)) |
3 | 2 | ssrdv 3574 | . 2 ⊢ (𝜑 → 𝐴 ⊆ ∅) |
4 | ss0 3926 | . 2 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅) | |
5 | 3, 4 | syl 17 | 1 ⊢ (𝜑 → 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 |
This theorem is referenced by: map0b 7782 disjen 8002 mapdom1 8010 pwxpndom2 9366 fzdisj 12239 smu01lem 15045 prmreclem5 15462 vdwap0 15518 natfval 16429 fucbas 16443 fuchom 16444 coafval 16537 efgval 17953 lsppratlem6 18973 lbsextlem4 18982 psrvscafval 19211 cfinufil 21542 ufinffr 21543 fin1aufil 21546 bldisj 22013 reconnlem1 22437 pcofval 22618 bcthlem5 22933 volfiniun 23122 fta1g 23731 fta1 23867 rpvmasum 25015 bj-projval 32177 finxpnom 32414 ipo0 37674 ifr0 37675 limclner 38718 |
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