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Mirrors > Home > MPE Home > Th. List > wdomen1 | Structured version Visualization version GIF version |
Description: Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.) |
Ref | Expression |
---|---|
wdomen1 | ⊢ (𝐴 ≈ 𝐵 → (𝐴 ≼* 𝐶 ↔ 𝐵 ≼* 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ensym 8544 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
2 | endom 8522 | . . . 4 ⊢ (𝐵 ≈ 𝐴 → 𝐵 ≼ 𝐴) | |
3 | domwdom 9024 | . . . 4 ⊢ (𝐵 ≼ 𝐴 → 𝐵 ≼* 𝐴) | |
4 | 1, 2, 3 | 3syl 18 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≼* 𝐴) |
5 | wdomtr 9025 | . . 3 ⊢ ((𝐵 ≼* 𝐴 ∧ 𝐴 ≼* 𝐶) → 𝐵 ≼* 𝐶) | |
6 | 4, 5 | sylan 582 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ≼* 𝐶) → 𝐵 ≼* 𝐶) |
7 | endom 8522 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
8 | domwdom 9024 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ≼* 𝐵) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼* 𝐵) |
10 | wdomtr 9025 | . . 3 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐵 ≼* 𝐶) → 𝐴 ≼* 𝐶) | |
11 | 9, 10 | sylan 582 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≼* 𝐶) → 𝐴 ≼* 𝐶) |
12 | 6, 11 | impbida 799 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ≼* 𝐶 ↔ 𝐵 ≼* 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 class class class wbr 5052 ≈ cen 8492 ≼ cdom 8493 ≼* cwdom 9007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-wdom 9009 |
This theorem is referenced by: (None) |
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