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Mirrors > Home > ILE Home > Th. List > rzal | GIF version |
Description: Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
rzal | ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0i 3430 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅) | |
2 | 1 | necon2bi 2402 | . . 3 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴) |
3 | 2 | pm2.21d 619 | . 2 ⊢ (𝐴 = ∅ → (𝑥 ∈ 𝐴 → 𝜑)) |
4 | 3 | ralrimiv 2549 | 1 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∅c0 3423 |
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-ne 2348 df-ral 2460 df-v 2740 df-dif 3132 df-nul 3424 |
This theorem is referenced by: ralf0 3527 fiubm 10808 mgm0 12788 sgrp0 12815 |
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