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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 2948 | . . . . . . 7 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) | |
2 | 1 | adantl 483 | . . . . . 6 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) |
3 | reldom 8895 | . . . . . . . . 9 ⊢ Rel ≼ | |
4 | 3 | brrelex1i 5692 | . . . . . . . 8 ⊢ (𝑋 ≼ 𝑌 → 𝑋 ∈ V) |
5 | 0sdomg 9054 | . . . . . . . 8 ⊢ (𝑋 ∈ V → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) | |
6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ (𝑋 ≼ 𝑌 → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) |
7 | 6 | adantr 482 | . . . . . 6 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) |
8 | 2, 7 | mpbird 257 | . . . . 5 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → ∅ ≺ 𝑋) |
9 | simpl 484 | . . . . 5 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → 𝑋 ≼ 𝑌) | |
10 | fodomr 9078 | . . . . 5 ⊢ ((∅ ≺ 𝑋 ∧ 𝑋 ≼ 𝑌) → ∃𝑦 𝑦:𝑌–onto→𝑋) | |
11 | 8, 9, 10 | syl2anc 585 | . . . 4 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → ∃𝑦 𝑦:𝑌–onto→𝑋) |
12 | 11 | ex 414 | . . 3 ⊢ (𝑋 ≼ 𝑌 → (¬ 𝑋 = ∅ → ∃𝑦 𝑦:𝑌–onto→𝑋)) |
13 | 12 | orrd 862 | . 2 ⊢ (𝑋 ≼ 𝑌 → (𝑋 = ∅ ∨ ∃𝑦 𝑦:𝑌–onto→𝑋)) |
14 | 3 | brrelex2i 5693 | . . 3 ⊢ (𝑋 ≼ 𝑌 → 𝑌 ∈ V) |
15 | brwdom 9511 | . . 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 257 | 1 ⊢ (𝑋 ≼ 𝑌 → 𝑋 ≼* 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2940 Vcvv 3447 ∅c0 4286 class class class wbr 5109 –onto→wfo 6498 ≼ cdom 8887 ≺ csdm 8888 ≼* cwdom 9508 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-en 8890 df-dom 8891 df-sdom 8892 df-wdom 9509 |
This theorem is referenced by: wdomen1 9520 wdomen2 9521 wdom2d 9524 wdomima2g 9530 unxpwdom2 9532 unxpwdom 9533 harwdom 9535 wdomfil 10005 wdomnumr 10008 pwdjudom 10160 hsmexlem1 10370 hsmexlem4 10373 |
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