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Mirrors > Home > ILE Home > Th. List > dom0 | GIF version |
Description: A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.) |
Ref | Expression |
---|---|
dom0 | ⊢ (𝐴 ≼ ∅ ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdomi 6706 | . . 3 ⊢ (𝐴 ≼ ∅ → ∃𝑥 𝑥:𝐴–1-1→∅) | |
2 | f1f 5387 | . . . . . 6 ⊢ (𝑥:𝐴–1-1→∅ → 𝑥:𝐴⟶∅) | |
3 | f00 5373 | . . . . . 6 ⊢ (𝑥:𝐴⟶∅ ↔ (𝑥 = ∅ ∧ 𝐴 = ∅)) | |
4 | 2, 3 | sylib 121 | . . . . 5 ⊢ (𝑥:𝐴–1-1→∅ → (𝑥 = ∅ ∧ 𝐴 = ∅)) |
5 | 4 | simprd 113 | . . . 4 ⊢ (𝑥:𝐴–1-1→∅ → 𝐴 = ∅) |
6 | 5 | adantl 275 | . . 3 ⊢ ((𝐴 ≼ ∅ ∧ 𝑥:𝐴–1-1→∅) → 𝐴 = ∅) |
7 | 1, 6 | exlimddv 1885 | . 2 ⊢ (𝐴 ≼ ∅ → 𝐴 = ∅) |
8 | 0ex 4103 | . . . 4 ⊢ ∅ ∈ V | |
9 | domrefg 6724 | . . . 4 ⊢ (∅ ∈ V → ∅ ≼ ∅) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ ∅ ≼ ∅ |
11 | breq1 3979 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≼ ∅ ↔ ∅ ≼ ∅)) | |
12 | 10, 11 | mpbiri 167 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≼ ∅) |
13 | 7, 12 | impbii 125 | 1 ⊢ (𝐴 ≼ ∅ ↔ 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 Vcvv 2721 ∅c0 3404 class class class wbr 3976 ⟶wf 5178 –1-1→wf1 5179 ≼ cdom 6696 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-en 6698 df-dom 6699 |
This theorem is referenced by: (None) |
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