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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 9139). (Contributed by BTernaryTau, 22-Nov-2024.) |
| Ref | Expression |
|---|---|
| domnsymfi | ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵) → ¬ 𝐵 ≺ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | brdom2 9022 | . 2 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) | |
| 2 | sdomnen 9021 | . . . . 5 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) | |
| 3 | 2 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≺ 𝐵) → ¬ 𝐴 ≈ 𝐵) |
| 4 | sdomdom 9020 | . . . . . . 7 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) | |
| 5 | sdomdom 9020 | . . . . . . . 8 ⊢ (𝐵 ≺ 𝐴 → 𝐵 ≼ 𝐴) | |
| 6 | sbthfi 9239 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴 ∧ 𝐴 ≼ 𝐵) → 𝐵 ≈ 𝐴) | |
| 7 | ensymfib 9224 | . . . . . . . . . 10 ⊢ (𝐴 ∈ Fin → (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴)) | |
| 8 | 7 | 3ad2ant1 1134 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴 ∧ 𝐴 ≼ 𝐵) → (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴)) |
| 9 | 6, 8 | mpbird 257 | . . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴 ∧ 𝐴 ≼ 𝐵) → 𝐴 ≈ 𝐵) |
| 10 | 5, 9 | syl3an2 1165 | . . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≺ 𝐴 ∧ 𝐴 ≼ 𝐵) → 𝐴 ≈ 𝐵) |
| 11 | 4, 10 | syl3an3 1166 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≺ 𝐴 ∧ 𝐴 ≺ 𝐵) → 𝐴 ≈ 𝐵) |
| 12 | 11 | 3com23 1127 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐴) → 𝐴 ≈ 𝐵) |
| 13 | 12 | 3expa 1119 | . . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ≺ 𝐵) ∧ 𝐵 ≺ 𝐴) → 𝐴 ≈ 𝐵) |
| 14 | 3, 13 | mtand 816 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≺ 𝐵) → ¬ 𝐵 ≺ 𝐴) |
| 15 | sdomnen 9021 | . . . 4 ⊢ (𝐵 ≺ 𝐴 → ¬ 𝐵 ≈ 𝐴) | |
| 16 | 7 | biimpa 476 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐵 ≈ 𝐴) |
| 17 | 15, 16 | nsyl3 138 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵) → ¬ 𝐵 ≺ 𝐴) |
| 18 | 14, 17 | jaodan 960 | . 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 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 ∈ wcel 2108 class class class wbr 5143 ≈ cen 8982 ≼ cdom 8983 ≺ csdm 8984 Fincfn 8985 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-1o 8506 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 |
| This theorem is referenced by: sdomdomtrfi 9241 domsdomtrfi 9242 nndomog 9253 onomeneq 9265 |
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