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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 3290 | . . 3 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1 | pm2.21i 610 | . 2 ⊢ (𝑥 ∈ ∅ → 𝜑) |
3 | 2 | rgen 2428 | 1 ⊢ ∀𝑥 ∈ ∅ 𝜑 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1438 ∀wral 2359 ∅c0 3286 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-v 2621 df-dif 3001 df-nul 3287 |
This theorem is referenced by: 0iin 3786 po0 4136 so0 4151 we0 4186 ord0 4216 omsinds 4433 mpt0 5135 iso0 5588 ac6sfi 6604 fimax2gtri 6607 finomni 6786 uzsinds 9836 seq3f1olemp 9919 rexfiuz 10410 fimaxre2 10645 2prm 11374 bj-nntrans 11729 |
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