![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > dm0 | GIF version |
Description: The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
dm0 | ⊢ dom ∅ = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eq0 3443 | . 2 ⊢ (dom ∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom ∅) | |
2 | noel 3428 | . . . 4 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
3 | 2 | nex 1500 | . . 3 ⊢ ¬ ∃𝑦〈𝑥, 𝑦〉 ∈ ∅ |
4 | vex 2742 | . . . 4 ⊢ 𝑥 ∈ V | |
5 | 4 | eldm2 4827 | . . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ ∅) |
6 | 3, 5 | mtbir 671 | . 2 ⊢ ¬ 𝑥 ∈ dom ∅ |
7 | 1, 6 | mpgbir 1453 | 1 ⊢ dom ∅ = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 = wceq 1353 ∃wex 1492 ∈ wcel 2148 ∅c0 3424 〈cop 3597 dom cdm 4628 |
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-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2741 df-dif 3133 df-un 3135 df-nul 3425 df-sn 3600 df-pr 3601 df-op 3603 df-br 4006 df-dm 4638 |
This theorem is referenced by: rn0 4885 sqxpeq0 5054 fn0 5337 f0dom0 5411 f1o00 5498 rdg0 6390 frec0g 6400 ennnfonelemj0 12404 ennnfonelem1 12410 ennnfonelemkh 12415 ennnfonelemhf1o 12416 |
Copyright terms: Public domain | W3C validator |