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Mirrors > Home > MPE Home > Th. List > domwdom | Structured version Visualization version GIF version |
Description: Weak dominance is implied by dominance in the usual sense. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
Ref | Expression |
---|---|
domwdom | ⊢ (𝑋 ≼ 𝑌 → 𝑋 ≼* 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neqne 2951 | . . . . . . 7 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) | |
2 | 1 | adantl 482 | . . . . . 6 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) |
3 | reldom 8739 | . . . . . . . . 9 ⊢ Rel ≼ | |
4 | 3 | brrelex1i 5643 | . . . . . . . 8 ⊢ (𝑋 ≼ 𝑌 → 𝑋 ∈ V) |
5 | 0sdomg 8891 | . . . . . . . 8 ⊢ (𝑋 ∈ V → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) | |
6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ (𝑋 ≼ 𝑌 → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) |
7 | 6 | adantr 481 | . . . . . 6 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) |
8 | 2, 7 | mpbird 256 | . . . . 5 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → ∅ ≺ 𝑋) |
9 | simpl 483 | . . . . 5 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → 𝑋 ≼ 𝑌) | |
10 | fodomr 8915 | . . . . 5 ⊢ ((∅ ≺ 𝑋 ∧ 𝑋 ≼ 𝑌) → ∃𝑦 𝑦:𝑌–onto→𝑋) | |
11 | 8, 9, 10 | syl2anc 584 | . . . 4 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → ∃𝑦 𝑦:𝑌–onto→𝑋) |
12 | 11 | ex 413 | . . 3 ⊢ (𝑋 ≼ 𝑌 → (¬ 𝑋 = ∅ → ∃𝑦 𝑦:𝑌–onto→𝑋)) |
13 | 12 | orrd 860 | . 2 ⊢ (𝑋 ≼ 𝑌 → (𝑋 = ∅ ∨ ∃𝑦 𝑦:𝑌–onto→𝑋)) |
14 | 3 | brrelex2i 5644 | . . 3 ⊢ (𝑋 ≼ 𝑌 → 𝑌 ∈ V) |
15 | brwdom 9326 | . . 3 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑦 𝑦:𝑌–onto→𝑋))) | |
16 | 14, 15 | syl 17 | . 2 ⊢ (𝑋 ≼ 𝑌 → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑦 𝑦:𝑌–onto→𝑋))) |
17 | 13, 16 | mpbird 256 | 1 ⊢ (𝑋 ≼ 𝑌 → 𝑋 ≼* 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ≠ wne 2943 Vcvv 3432 ∅c0 4256 class class class wbr 5074 –onto→wfo 6431 ≼ cdom 8731 ≺ csdm 8732 ≼* cwdom 9323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-en 8734 df-dom 8735 df-sdom 8736 df-wdom 9324 |
This theorem is referenced by: wdomen1 9335 wdomen2 9336 wdom2d 9339 wdomima2g 9345 unxpwdom2 9347 unxpwdom 9348 harwdom 9350 wdomfil 9817 wdomnumr 9820 pwdjudom 9972 hsmexlem1 10182 hsmexlem4 10185 |
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