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| Mirrors > Home > MPE Home > Th. List > fimaxg | Structured version Visualization version GIF version | ||
| Description: A finite set has a maximum under a total order. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.) | 
| Ref | Expression | 
|---|---|
| fimaxg | ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦𝑅𝑥)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fimax2g 9322 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦) | |
| 2 | df-ne 2941 | . . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦) | |
| 3 | 2 | imbi1i 349 | . . . . . . . 8 ⊢ ((𝑥 ≠ 𝑦 → 𝑦𝑅𝑥) ↔ (¬ 𝑥 = 𝑦 → 𝑦𝑅𝑥)) | 
| 4 | pm4.64 850 | . . . . . . . 8 ⊢ ((¬ 𝑥 = 𝑦 → 𝑦𝑅𝑥) ↔ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) | |
| 5 | 3, 4 | bitri 275 | . . . . . . 7 ⊢ ((𝑥 ≠ 𝑦 → 𝑦𝑅𝑥) ↔ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) | 
| 6 | sotric 5622 | . . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | |
| 7 | 6 | con2bid 354 | . . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦)) | 
| 8 | 5, 7 | bitrid 283 | . . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥 ≠ 𝑦 → 𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦)) | 
| 9 | 8 | anassrs 467 | . . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝑥 ≠ 𝑦 → 𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦)) | 
| 10 | 9 | ralbidva 3176 | . . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦)) | 
| 11 | 10 | rexbidva 3177 | . . 3 ⊢ (𝑅 Or 𝐴 → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦𝑅𝑥) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦)) | 
| 12 | 11 | 3ad2ant1 1134 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦𝑅𝑥) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦)) | 
| 13 | 1, 12 | mpbird 257 | 1 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑦𝑅𝑥)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ∅c0 4333 class class class wbr 5143 Or wor 5591 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-en 8986 df-fin 8989 | 
| This theorem is referenced by: fisupg 9324 fimaxre 12212 | 
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