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Mirrors > Home > MPE Home > Th. List > fiming | Structured version Visualization version GIF version |
Description: A finite set has a minimum under a total order. (Contributed by AV, 6-Oct-2020.) |
Ref | Expression |
---|---|
fiming | ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑥𝑅𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fimin2g 8286 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) | |
2 | nesym 2838 | . . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑦 = 𝑥) | |
3 | 2 | imbi1i 338 | . . . . . . . 8 ⊢ ((𝑥 ≠ 𝑦 → 𝑥𝑅𝑦) ↔ (¬ 𝑦 = 𝑥 → 𝑥𝑅𝑦)) |
4 | pm4.64 386 | . . . . . . . 8 ⊢ ((¬ 𝑦 = 𝑥 → 𝑥𝑅𝑦) ↔ (𝑦 = 𝑥 ∨ 𝑥𝑅𝑦)) | |
5 | 3, 4 | bitri 263 | . . . . . . 7 ⊢ ((𝑥 ≠ 𝑦 → 𝑥𝑅𝑦) ↔ (𝑦 = 𝑥 ∨ 𝑥𝑅𝑦)) |
6 | sotric 4985 | . . . . . . . . 9 ⊢ ((𝑅 Or 𝐴 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝑦𝑅𝑥 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑥𝑅𝑦))) | |
7 | 6 | ancom2s 840 | . . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑦𝑅𝑥 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑥𝑅𝑦))) |
8 | 7 | con2bid 343 | . . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑦 = 𝑥 ∨ 𝑥𝑅𝑦) ↔ ¬ 𝑦𝑅𝑥)) |
9 | 5, 8 | syl5bb 271 | . . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥 ≠ 𝑦 → 𝑥𝑅𝑦) ↔ ¬ 𝑦𝑅𝑥)) |
10 | 9 | anassrs 678 | . . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝑥 ≠ 𝑦 → 𝑥𝑅𝑦) ↔ ¬ 𝑦𝑅𝑥)) |
11 | 10 | ralbidva 2968 | . . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)) |
12 | 11 | rexbidva 3031 | . . 3 ⊢ (𝑅 Or 𝐴 → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑥𝑅𝑦) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)) |
13 | 12 | 3ad2ant1 1075 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑥𝑅𝑦) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)) |
14 | 1, 13 | mpbird 246 | 1 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑥𝑅𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ∅c0 3874 class class class wbr 4583 Or wor 4958 Fincfn 7841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-1o 7447 df-er 7629 df-en 7842 df-fin 7845 |
This theorem is referenced by: fiinfg 8288 |
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