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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 6748 | . . 3 ⊢ (𝐴 ≼ ∅ → ∃𝑥 𝑥:𝐴–1-1→∅) | |
2 | f1f 5421 | . . . . . 6 ⊢ (𝑥:𝐴–1-1→∅ → 𝑥:𝐴⟶∅) | |
3 | f00 5407 | . . . . . 6 ⊢ (𝑥:𝐴⟶∅ ↔ (𝑥 = ∅ ∧ 𝐴 = ∅)) | |
4 | 2, 3 | sylib 122 | . . . . 5 ⊢ (𝑥:𝐴–1-1→∅ → (𝑥 = ∅ ∧ 𝐴 = ∅)) |
5 | 4 | simprd 114 | . . . 4 ⊢ (𝑥:𝐴–1-1→∅ → 𝐴 = ∅) |
6 | 5 | adantl 277 | . . 3 ⊢ ((𝐴 ≼ ∅ ∧ 𝑥:𝐴–1-1→∅) → 𝐴 = ∅) |
7 | 1, 6 | exlimddv 1898 | . 2 ⊢ (𝐴 ≼ ∅ → 𝐴 = ∅) |
8 | 0ex 4130 | . . . 4 ⊢ ∅ ∈ V | |
9 | domrefg 6766 | . . . 4 ⊢ (∅ ∈ V → ∅ ≼ ∅) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ ∅ ≼ ∅ |
11 | breq1 4006 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≼ ∅ ↔ ∅ ≼ ∅)) | |
12 | 10, 11 | mpbiri 168 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≼ ∅) |
13 | 7, 12 | impbii 126 | 1 ⊢ (𝐴 ≼ ∅ ↔ 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ∅c0 3422 class class class wbr 4003 ⟶wf 5212 –1-1→wf1 5213 ≼ cdom 6738 |
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-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-en 6740 df-dom 6741 |
This theorem is referenced by: (None) |
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