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Mirrors > Home > MPE Home > Th. List > domnsymfi | Structured version Visualization version GIF version |
Description: If a set dominates a finite set, it cannot also be strictly dominated by the finite set. This theorem is proved without using the Axiom of Power Sets (unlike domnsym 8877). (Contributed by BTernaryTau, 22-Nov-2024.) |
Ref | Expression |
---|---|
domnsymfi | ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵) → ¬ 𝐵 ≺ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdom2 8762 | . 2 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) | |
2 | sdomnen 8761 | . . . . 5 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) | |
3 | 2 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≺ 𝐵) → ¬ 𝐴 ≈ 𝐵) |
4 | sdomdom 8760 | . . . . . . 7 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) | |
5 | sdomdom 8760 | . . . . . . . 8 ⊢ (𝐵 ≺ 𝐴 → 𝐵 ≼ 𝐴) | |
6 | sbthfi 8976 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴 ∧ 𝐴 ≼ 𝐵) → 𝐵 ≈ 𝐴) | |
7 | ensymfib 8961 | . . . . . . . . . 10 ⊢ (𝐴 ∈ Fin → (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴)) | |
8 | 7 | 3ad2ant1 1132 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴 ∧ 𝐴 ≼ 𝐵) → (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴)) |
9 | 6, 8 | mpbird 256 | . . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴 ∧ 𝐴 ≼ 𝐵) → 𝐴 ≈ 𝐵) |
10 | 5, 9 | syl3an2 1163 | . . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≺ 𝐴 ∧ 𝐴 ≼ 𝐵) → 𝐴 ≈ 𝐵) |
11 | 4, 10 | syl3an3 1164 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≺ 𝐴 ∧ 𝐴 ≺ 𝐵) → 𝐴 ≈ 𝐵) |
12 | 11 | 3com23 1125 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐴) → 𝐴 ≈ 𝐵) |
13 | 12 | 3expa 1117 | . . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ≺ 𝐵) ∧ 𝐵 ≺ 𝐴) → 𝐴 ≈ 𝐵) |
14 | 3, 13 | mtand 813 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≺ 𝐵) → ¬ 𝐵 ≺ 𝐴) |
15 | sdomnen 8761 | . . . 4 ⊢ (𝐵 ≺ 𝐴 → ¬ 𝐵 ≈ 𝐴) | |
16 | 7 | biimpa 477 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐵 ≈ 𝐴) |
17 | 15, 16 | nsyl3 138 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵) → ¬ 𝐵 ≺ 𝐴) |
18 | 14, 17 | jaodan 955 | . 2 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) → ¬ 𝐵 ≺ 𝐴) |
19 | 1, 18 | sylan2b 594 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵) → ¬ 𝐵 ≺ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 ∧ 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: sdomdomtrfi 8978 domsdomtrfi 8979 nndomog 8989 |
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