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Mirrors > Home > MPE Home > Th. List > wdompwdom | Structured version Visualization version GIF version |
Description: Weak dominance strengthens to usual dominance on the power sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
Ref | Expression |
---|---|
wdompwdom | ⊢ (𝑋 ≼* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relwdom 8354 | . . . . . 6 ⊢ Rel ≼* | |
2 | 1 | brrelex2i 5083 | . . . . 5 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V) |
3 | pwexg 4776 | . . . . 5 ⊢ (𝑌 ∈ V → 𝒫 𝑌 ∈ V) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝑋 ≼* 𝑌 → 𝒫 𝑌 ∈ V) |
5 | 0ss 3924 | . . . . 5 ⊢ ∅ ⊆ 𝑌 | |
6 | sspwb 4844 | . . . . 5 ⊢ (∅ ⊆ 𝑌 ↔ 𝒫 ∅ ⊆ 𝒫 𝑌) | |
7 | 5, 6 | mpbi 219 | . . . 4 ⊢ 𝒫 ∅ ⊆ 𝒫 𝑌 |
8 | ssdomg 7887 | . . . 4 ⊢ (𝒫 𝑌 ∈ V → (𝒫 ∅ ⊆ 𝒫 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌)) | |
9 | 4, 7, 8 | mpisyl 21 | . . 3 ⊢ (𝑋 ≼* 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌) |
10 | pweq 4111 | . . . 4 ⊢ (𝑋 = ∅ → 𝒫 𝑋 = 𝒫 ∅) | |
11 | 10 | breq1d 4593 | . . 3 ⊢ (𝑋 = ∅ → (𝒫 𝑋 ≼ 𝒫 𝑌 ↔ 𝒫 ∅ ≼ 𝒫 𝑌)) |
12 | 9, 11 | syl5ibr 235 | . 2 ⊢ (𝑋 = ∅ → (𝑋 ≼* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)) |
13 | brwdomn0 8357 | . . 3 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)) | |
14 | vex 3176 | . . . . 5 ⊢ 𝑧 ∈ V | |
15 | fopwdom 7953 | . . . . 5 ⊢ ((𝑧 ∈ V ∧ 𝑧:𝑌–onto→𝑋) → 𝒫 𝑋 ≼ 𝒫 𝑌) | |
16 | 14, 15 | mpan 702 | . . . 4 ⊢ (𝑧:𝑌–onto→𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌) |
17 | 16 | exlimiv 1845 | . . 3 ⊢ (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌) |
18 | 13, 17 | syl6bi 242 | . 2 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)) |
19 | 12, 18 | pm2.61ine 2865 | 1 ⊢ (𝑋 ≼* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 class class class wbr 4583 –onto→wfo 5802 ≼ cdom 7839 ≼* cwdom 8345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-dom 7843 df-wdom 8347 |
This theorem is referenced by: isfin32i 9070 hsmexlem1 9131 hsmexlem3 9133 gchhar 9380 |
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