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Mirrors > Home > MPE Home > Th. List > fidomtri2 | Structured version Visualization version GIF version |
Description: Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 7-May-2015.) |
Ref | Expression |
---|---|
fidomtri2 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domnsym 7971 | . 2 ⊢ (𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴) | |
2 | sdomdom 7869 | . . . . . . 7 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) | |
3 | 2 | con3i 149 | . . . . . 6 ⊢ (¬ 𝐴 ≼ 𝐵 → ¬ 𝐴 ≺ 𝐵) |
4 | fidomtri 8702 | . . . . . . 7 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ∈ 𝑉) → (𝐵 ≼ 𝐴 ↔ ¬ 𝐴 ≺ 𝐵)) | |
5 | 4 | ancoms 468 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin) → (𝐵 ≼ 𝐴 ↔ ¬ 𝐴 ≺ 𝐵)) |
6 | 3, 5 | syl5ibr 235 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin) → (¬ 𝐴 ≼ 𝐵 → 𝐵 ≼ 𝐴)) |
7 | ensym 7891 | . . . . . . . 8 ⊢ (𝐵 ≈ 𝐴 → 𝐴 ≈ 𝐵) | |
8 | endom 7868 | . . . . . . . 8 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
9 | 7, 8 | syl 17 | . . . . . . 7 ⊢ (𝐵 ≈ 𝐴 → 𝐴 ≼ 𝐵) |
10 | 9 | con3i 149 | . . . . . 6 ⊢ (¬ 𝐴 ≼ 𝐵 → ¬ 𝐵 ≈ 𝐴) |
11 | 10 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin) → (¬ 𝐴 ≼ 𝐵 → ¬ 𝐵 ≈ 𝐴)) |
12 | 6, 11 | jcad 554 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin) → (¬ 𝐴 ≼ 𝐵 → (𝐵 ≼ 𝐴 ∧ ¬ 𝐵 ≈ 𝐴))) |
13 | brsdom 7864 | . . . 4 ⊢ (𝐵 ≺ 𝐴 ↔ (𝐵 ≼ 𝐴 ∧ ¬ 𝐵 ≈ 𝐴)) | |
14 | 12, 13 | syl6ibr 241 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin) → (¬ 𝐴 ≼ 𝐵 → 𝐵 ≺ 𝐴)) |
15 | 14 | con1d 138 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin) → (¬ 𝐵 ≺ 𝐴 → 𝐴 ≼ 𝐵)) |
16 | 1, 15 | impbid2 215 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 class class class wbr 4583 ≈ cen 7838 ≼ cdom 7839 ≺ csdm 7840 Fincfn 7841 |
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-3or 1032 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-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 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-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 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-om 6958 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 |
This theorem is referenced by: gchdomtri 9330 gchcda1 9357 frgpcyg 19741 |
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