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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 9284 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦) | |
2 | df-ne 2942 | . . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦) | |
3 | 2 | imbi1i 350 | . . . . . . . 8 ⊢ ((𝑥 ≠ 𝑦 → 𝑦𝑅𝑥) ↔ (¬ 𝑥 = 𝑦 → 𝑦𝑅𝑥)) |
4 | pm4.64 848 | . . . . . . . 8 ⊢ ((¬ 𝑥 = 𝑦 → 𝑦𝑅𝑥) ↔ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) | |
5 | 3, 4 | bitri 275 | . . . . . . 7 ⊢ ((𝑥 ≠ 𝑦 → 𝑦𝑅𝑥) ↔ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) |
6 | sotric 5614 | . . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | |
7 | 6 | con2bid 355 | . . . . . . 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 469 | . . . . 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 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 ∅c0 4320 class class class wbr 5146 Or wor 5585 Fincfn 8934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pr 5425 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-br 5147 df-opab 5209 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-om 7850 df-en 8935 df-fin 8938 |
This theorem is referenced by: fisupg 9286 fimaxre 12153 |
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