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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 3432 | . 2 ⊢ (dom ∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom ∅) | |
2 | noel 3418 | . . . 4 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
3 | 2 | nex 1493 | . . 3 ⊢ ¬ ∃𝑦〈𝑥, 𝑦〉 ∈ ∅ |
4 | vex 2733 | . . . 4 ⊢ 𝑥 ∈ V | |
5 | 4 | eldm2 4807 | . . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ ∅) |
6 | 3, 5 | mtbir 666 | . 2 ⊢ ¬ 𝑥 ∈ dom ∅ |
7 | 1, 6 | mpgbir 1446 | 1 ⊢ dom ∅ = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 = wceq 1348 ∃wex 1485 ∈ wcel 2141 ∅c0 3414 〈cop 3584 dom cdm 4609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-un 3125 df-nul 3415 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-dm 4619 |
This theorem is referenced by: rn0 4865 sqxpeq0 5032 fn0 5315 f0dom0 5389 f1o00 5475 rdg0 6363 frec0g 6373 ennnfonelemj0 12343 ennnfonelem1 12349 ennnfonelemkh 12354 ennnfonelemhf1o 12355 |
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