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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 | df-ne 3000 | . . . . . . . 8 ⊢ (𝑋 ≠ ∅ ↔ ¬ 𝑋 = ∅) | |
2 | 1 | biimpri 220 | . . . . . . 7 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) |
3 | 2 | adantl 475 | . . . . . 6 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) |
4 | reldom 8228 | . . . . . . . . 9 ⊢ Rel ≼ | |
5 | 4 | brrelex1i 5393 | . . . . . . . 8 ⊢ (𝑋 ≼ 𝑌 → 𝑋 ∈ V) |
6 | 0sdomg 8358 | . . . . . . . 8 ⊢ (𝑋 ∈ V → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (𝑋 ≼ 𝑌 → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) |
8 | 7 | adantr 474 | . . . . . 6 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) |
9 | 3, 8 | mpbird 249 | . . . . 5 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → ∅ ≺ 𝑋) |
10 | simpl 476 | . . . . 5 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → 𝑋 ≼ 𝑌) | |
11 | fodomr 8380 | . . . . 5 ⊢ ((∅ ≺ 𝑋 ∧ 𝑋 ≼ 𝑌) → ∃𝑦 𝑦:𝑌–onto→𝑋) | |
12 | 9, 10, 11 | syl2anc 579 | . . . 4 ⊢ ((𝑋 ≼ 𝑌 ∧ ¬ 𝑋 = ∅) → ∃𝑦 𝑦:𝑌–onto→𝑋) |
13 | 12 | ex 403 | . . 3 ⊢ (𝑋 ≼ 𝑌 → (¬ 𝑋 = ∅ → ∃𝑦 𝑦:𝑌–onto→𝑋)) |
14 | 13 | orrd 894 | . 2 ⊢ (𝑋 ≼ 𝑌 → (𝑋 = ∅ ∨ ∃𝑦 𝑦:𝑌–onto→𝑋)) |
15 | 4 | brrelex2i 5394 | . . 3 ⊢ (𝑋 ≼ 𝑌 → 𝑌 ∈ V) |
16 | brwdom 8741 | . . 3 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑦 𝑦:𝑌–onto→𝑋))) | |
17 | 15, 16 | syl 17 | . 2 ⊢ (𝑋 ≼ 𝑌 → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑦 𝑦:𝑌–onto→𝑋))) |
18 | 14, 17 | mpbird 249 | 1 ⊢ (𝑋 ≼ 𝑌 → 𝑋 ≼* 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∨ wo 878 = wceq 1656 ∃wex 1878 ∈ wcel 2164 ≠ wne 2999 Vcvv 3414 ∅c0 4144 class class class wbr 4873 –onto→wfo 6121 ≼ cdom 8220 ≺ csdm 8221 ≼* cwdom 8731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-wdom 8733 |
This theorem is referenced by: wdomen1 8750 wdomen2 8751 wdom2d 8754 wdomima2g 8760 unxpwdom2 8762 unxpwdom 8763 harwdom 8764 wdomfil 9197 wdomnumr 9200 pwcdadom 9353 hsmexlem1 9563 hsmexlem4 9566 |
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