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Mirrors > Home > ILE Home > Th. List > ral0 | GIF version |
Description: Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.) |
Ref | Expression |
---|---|
ral0 | ⊢ ∀𝑥 ∈ ∅ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3291 | . . 3 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1 | pm2.21i 611 | . 2 ⊢ (𝑥 ∈ ∅ → 𝜑) |
3 | 2 | rgen 2429 | 1 ⊢ ∀𝑥 ∈ ∅ 𝜑 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1439 ∀wral 2360 ∅c0 3287 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-v 2622 df-dif 3002 df-nul 3288 |
This theorem is referenced by: 0iin 3794 po0 4147 so0 4162 we0 4197 ord0 4227 omsinds 4448 mpt0 5154 iso0 5610 ixp0x 6497 ac6sfi 6668 fimax2gtri 6671 finomni 6857 uzsinds 9909 seq3f1olemp 9992 rexfiuz 10483 fimaxre2 10719 2prm 11448 bj-nntrans 12119 |
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