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Mirrors > Home > ILE Home > Th. List > ralf0 | GIF version |
Description: The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) |
Ref | Expression |
---|---|
ralf0.1 | ⊢ ¬ 𝜑 |
Ref | Expression |
---|---|
ralf0 | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralf0.1 | . . . . 5 ⊢ ¬ 𝜑 | |
2 | con3 571 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (¬ 𝜑 → ¬ 𝑥 ∈ 𝐴)) | |
3 | 1, 2 | mpi 15 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → ¬ 𝑥 ∈ 𝐴) |
4 | 3 | alimi 1344 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
5 | df-ral 2311 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
6 | eq0 3239 | . . 3 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | |
7 | 4, 5, 6 | 3imtr4i 190 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → 𝐴 = ∅) |
8 | rzal 3318 | . 2 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) | |
9 | 7, 8 | impbii 117 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 ∀wal 1241 = wceq 1243 ∈ wcel 1393 ∀wral 2306 ∅c0 3224 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-v 2559 df-dif 2920 df-nul 3225 |
This theorem is referenced by: (None) |
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