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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 8546 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
2 | endom 8524 | . . . 4 ⊢ (𝐵 ≈ 𝐴 → 𝐵 ≼ 𝐴) | |
3 | domwdom 9026 | . . . 4 ⊢ (𝐵 ≼ 𝐴 → 𝐵 ≼* 𝐴) | |
4 | 1, 2, 3 | 3syl 18 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≼* 𝐴) |
5 | wdomtr 9027 | . . 3 ⊢ ((𝐵 ≼* 𝐴 ∧ 𝐴 ≼* 𝐶) → 𝐵 ≼* 𝐶) | |
6 | 4, 5 | sylan 580 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ≼* 𝐶) → 𝐵 ≼* 𝐶) |
7 | endom 8524 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
8 | domwdom 9026 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ≼* 𝐵) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼* 𝐵) |
10 | wdomtr 9027 | . . 3 ⊢ ((𝐴 ≼* 𝐵 ∧ 𝐵 ≼* 𝐶) → 𝐴 ≼* 𝐶) | |
11 | 9, 10 | sylan 580 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≼* 𝐶) → 𝐴 ≼* 𝐶) |
12 | 6, 11 | impbida 797 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ≼* 𝐶 ↔ 𝐵 ≼* 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 class class class wbr 5057 ≈ cen 8494 ≼ cdom 8495 ≼* cwdom 9009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-wdom 9011 |
This theorem is referenced by: (None) |
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