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Mirrors > Home > ILE Home > Th. List > eq0rdv | GIF version |
Description: Deduction 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 619 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∅)) |
3 | 2 | ssrdv 3161 | . 2 ⊢ (𝜑 → 𝐴 ⊆ ∅) |
4 | ss0 3463 | . 2 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅) | |
5 | 3, 4 | syl 14 | 1 ⊢ (𝜑 → 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1353 ∈ wcel 2148 ⊆ wss 3129 ∅c0 3422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-dif 3131 df-in 3135 df-ss 3142 df-nul 3423 |
This theorem is referenced by: exmid01 4195 dcextest 4577 nfvres 5544 map0b 6681 snon0 6929 snexxph 6943 fodju0 7139 fzdisj 10035 bldisj 13565 |
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