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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 6643 | . . 3 ⊢ (𝐴 ≼ ∅ → ∃𝑥 𝑥:𝐴–1-1→∅) | |
2 | f1f 5328 | . . . . . 6 ⊢ (𝑥:𝐴–1-1→∅ → 𝑥:𝐴⟶∅) | |
3 | f00 5314 | . . . . . 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 1870 | . 2 ⊢ (𝐴 ≼ ∅ → 𝐴 = ∅) |
8 | 0ex 4055 | . . . 4 ⊢ ∅ ∈ V | |
9 | domrefg 6661 | . . . 4 ⊢ (∅ ∈ V → ∅ ≼ ∅) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ ∅ ≼ ∅ |
11 | breq1 3932 | . . 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 1331 ∈ wcel 1480 Vcvv 2686 ∅c0 3363 class class class wbr 3929 ⟶wf 5119 –1-1→wf1 5120 ≼ cdom 6633 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-en 6635 df-dom 6636 |
This theorem is referenced by: (None) |
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