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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 8819 | . . . . . 6 ⊢ Rel ≼* | |
2 | 1 | brrelex2i 5453 | . . . . 5 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V) |
3 | 2 | pwexd 5127 | . . . 4 ⊢ (𝑋 ≼* 𝑌 → 𝒫 𝑌 ∈ V) |
4 | 0ss 4230 | . . . . 5 ⊢ ∅ ⊆ 𝑌 | |
5 | sspwb 5192 | . . . . 5 ⊢ (∅ ⊆ 𝑌 ↔ 𝒫 ∅ ⊆ 𝒫 𝑌) | |
6 | 4, 5 | mpbi 222 | . . . 4 ⊢ 𝒫 ∅ ⊆ 𝒫 𝑌 |
7 | ssdomg 8346 | . . . 4 ⊢ (𝒫 𝑌 ∈ V → (𝒫 ∅ ⊆ 𝒫 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌)) | |
8 | 3, 6, 7 | mpisyl 21 | . . 3 ⊢ (𝑋 ≼* 𝑌 → 𝒫 ∅ ≼ 𝒫 𝑌) |
9 | pweq 4419 | . . . 4 ⊢ (𝑋 = ∅ → 𝒫 𝑋 = 𝒫 ∅) | |
10 | 9 | breq1d 4933 | . . 3 ⊢ (𝑋 = ∅ → (𝒫 𝑋 ≼ 𝒫 𝑌 ↔ 𝒫 ∅ ≼ 𝒫 𝑌)) |
11 | 8, 10 | syl5ibr 238 | . 2 ⊢ (𝑋 = ∅ → (𝑋 ≼* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)) |
12 | brwdomn0 8822 | . . 3 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)) | |
13 | vex 3412 | . . . . 5 ⊢ 𝑧 ∈ V | |
14 | fopwdom 8415 | . . . . 5 ⊢ ((𝑧 ∈ V ∧ 𝑧:𝑌–onto→𝑋) → 𝒫 𝑋 ≼ 𝒫 𝑌) | |
15 | 13, 14 | mpan 677 | . . . 4 ⊢ (𝑧:𝑌–onto→𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌) |
16 | 15 | exlimiv 1889 | . . 3 ⊢ (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝒫 𝑋 ≼ 𝒫 𝑌) |
17 | 12, 16 | syl6bi 245 | . 2 ⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)) |
18 | 11, 17 | pm2.61ine 3045 | 1 ⊢ (𝑋 ≼* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∃wex 1742 ∈ wcel 2050 ≠ wne 2961 Vcvv 3409 ⊆ wss 3823 ∅c0 4172 𝒫 cpw 4416 class class class wbr 4923 –onto→wfo 6180 ≼ cdom 8298 ≼* cwdom 8810 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5306 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-dom 8302 df-wdom 8812 |
This theorem is referenced by: isfin32i 9579 hsmexlem1 9640 hsmexlem3 9642 gchhar 9893 |
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