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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 2995 | . . . . . . 7 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) | |
2 | 1 | adantl 485 | . . . . . 6 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) |
3 | reldom 8498 | . . . . . . . . 9 ⊢ Rel ≼ | |
4 | 3 | brrelex1i 5572 | . . . . . . . 8 ⊢ (𝑋 ≼ 𝑌 → 𝑋 ∈ V) |
5 | 0sdomg 8630 | . . . . . . . 8 ⊢ (𝑋 ∈ V → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) | |
6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ (𝑋 ≼ 𝑌 → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) |
7 | 6 | adantr 484 | . . . . . 6 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) |
8 | 2, 7 | mpbird 260 | . . . . 5 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → ∅ ≺ 𝑋) |
9 | simpl 486 | . . . . 5 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → 𝑋 ≼ 𝑌) | |
10 | fodomr 8652 | . . . . 5 ⊢ ((∅ ≺ 𝑋 ∧ 𝑋 ≼ 𝑌) → ∃𝑦 𝑦:𝑌–onto→𝑋) | |
11 | 8, 9, 10 | syl2anc 587 | . . . 4 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → ∃𝑦 𝑦:𝑌–onto→𝑋) |
12 | 11 | ex 416 | . . 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 5573 | . . 3 ⊢ (𝑋 ≼ 𝑌 → 𝑌 ∈ V) |
15 | brwdom 9015 | . . 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 260 | 1 ⊢ (𝑋 ≼ 𝑌 → 𝑋 ≼* 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∅c0 4243 class class class wbr 5030 –onto→wfo 6322 ≼ cdom 8490 ≺ csdm 8491 ≼* cwdom 9012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-wdom 9013 |
This theorem is referenced by: wdomen1 9024 wdomen2 9025 wdom2d 9028 wdomima2g 9034 unxpwdom2 9036 unxpwdom 9037 harwdom 9039 wdomfil 9472 wdomnumr 9475 pwdjudom 9627 hsmexlem1 9837 hsmexlem4 9840 |
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