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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 3441 | . . 3 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1 | pm2.21i 647 | . 2 ⊢ (𝑥 ∈ ∅ → 𝜑) |
3 | 2 | rgen 2543 | 1 ⊢ ∀𝑥 ∈ ∅ 𝜑 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 ∀wral 2468 ∅c0 3437 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2754 df-dif 3146 df-nul 3438 |
This theorem is referenced by: 0iin 3960 po0 4329 so0 4344 we0 4379 ord0 4409 omsinds 4639 mpt0 5362 iso0 5839 ixp0x 6752 ac6sfi 6926 fimax2gtri 6929 dcfi 7010 nnnninfeq2 7157 nninfisollem0 7158 finomni 7168 uzsinds 10473 seq3f1olemp 10533 rexfiuz 11030 fimaxre2 11267 2prm 12159 bj-nntrans 15161 |
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