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Mirrors > Home > MPE Home > Th. List > domsdomtrfi | Structured version Visualization version GIF version |
Description: Transitivity of dominance and strict dominance when 𝐴 is finite, proved without using the Axiom of Power Sets (unlike domsdomtr 8890). (Contributed by BTernaryTau, 25-Nov-2024.) |
Ref | Expression |
---|---|
domsdomtrfi | ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdomdom 8760 | . . 3 ⊢ (𝐵 ≺ 𝐶 → 𝐵 ≼ 𝐶) | |
2 | domtrfil 8969 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | |
3 | 1, 2 | syl3an3 1164 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≼ 𝐶) |
4 | ensymfib 8961 | . . . . . . . . . 10 ⊢ (𝐴 ∈ Fin → (𝐴 ≈ 𝐶 ↔ 𝐶 ≈ 𝐴)) | |
5 | 4 | biimpa 477 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐶) → 𝐶 ≈ 𝐴) |
6 | 5 | 3adant3 1131 | . . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐶 ∧ 𝐴 ≼ 𝐵) → 𝐶 ≈ 𝐴) |
7 | enfii 8963 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ≈ 𝐴) → 𝐶 ∈ Fin) | |
8 | 7 | 3adant3 1131 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ≈ 𝐴 ∧ 𝐴 ≼ 𝐵) → 𝐶 ∈ Fin) |
9 | endom 8759 | . . . . . . . . . 10 ⊢ (𝐶 ≈ 𝐴 → 𝐶 ≼ 𝐴) | |
10 | domtrfi 8970 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ≼ 𝐴 ∧ 𝐴 ≼ 𝐵) → 𝐶 ≼ 𝐵) | |
11 | 9, 10 | syl3an2 1163 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ≈ 𝐴 ∧ 𝐴 ≼ 𝐵) → 𝐶 ≼ 𝐵) |
12 | 8, 11 | jca 512 | . . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ≈ 𝐴 ∧ 𝐴 ≼ 𝐵) → (𝐶 ∈ Fin ∧ 𝐶 ≼ 𝐵)) |
13 | 6, 12 | syld3an2 1410 | . . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐶 ∧ 𝐴 ≼ 𝐵) → (𝐶 ∈ Fin ∧ 𝐶 ≼ 𝐵)) |
14 | domnsymfi 8977 | . . . . . . 7 ⊢ ((𝐶 ∈ Fin ∧ 𝐶 ≼ 𝐵) → ¬ 𝐵 ≺ 𝐶) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐶 ∧ 𝐴 ≼ 𝐵) → ¬ 𝐵 ≺ 𝐶) |
16 | 15 | 3com23 1125 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐴 ≈ 𝐶) → ¬ 𝐵 ≺ 𝐶) |
17 | 16 | 3expia 1120 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵) → (𝐴 ≈ 𝐶 → ¬ 𝐵 ≺ 𝐶)) |
18 | 17 | con2d 134 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵) → (𝐵 ≺ 𝐶 → ¬ 𝐴 ≈ 𝐶)) |
19 | 18 | 3impia 1116 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → ¬ 𝐴 ≈ 𝐶) |
20 | brsdom 8755 | . 2 ⊢ (𝐴 ≺ 𝐶 ↔ (𝐴 ≼ 𝐶 ∧ ¬ 𝐴 ≈ 𝐶)) | |
21 | 3, 19, 20 | sylanbrc 583 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 ∈ wcel 2110 class class class wbr 5079 ≈ cen 8722 ≼ cdom 8723 ≺ csdm 8724 Fincfn 8725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-un 7583 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-om 7708 df-1o 8289 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 |
This theorem is referenced by: php3 8985 |
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