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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 8961 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) | |
2 | nesym 3072 | . . . . . . . . 9 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑦 = 𝑥) | |
3 | 2 | imbi1i 352 | . . . . . . . 8 ⊢ ((𝑥 ≠ 𝑦 → 𝑥𝑅𝑦) ↔ (¬ 𝑦 = 𝑥 → 𝑥𝑅𝑦)) |
4 | pm4.64 845 | . . . . . . . 8 ⊢ ((¬ 𝑦 = 𝑥 → 𝑥𝑅𝑦) ↔ (𝑦 = 𝑥 ∨ 𝑥𝑅𝑦)) | |
5 | 3, 4 | bitri 277 | . . . . . . 7 ⊢ ((𝑥 ≠ 𝑦 → 𝑥𝑅𝑦) ↔ (𝑦 = 𝑥 ∨ 𝑥𝑅𝑦)) |
6 | sotric 5501 | . . . . . . . . 9 ⊢ ((𝑅 Or 𝐴 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝑦𝑅𝑥 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑥𝑅𝑦))) | |
7 | 6 | ancom2s 648 | . . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑦𝑅𝑥 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑥𝑅𝑦))) |
8 | 7 | con2bid 357 | . . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑦 = 𝑥 ∨ 𝑥𝑅𝑦) ↔ ¬ 𝑦𝑅𝑥)) |
9 | 5, 8 | syl5bb 285 | . . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥 ≠ 𝑦 → 𝑥𝑅𝑦) ↔ ¬ 𝑦𝑅𝑥)) |
10 | 9 | anassrs 470 | . . . . 5 ⊢ (((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ((𝑥 ≠ 𝑦 → 𝑥𝑅𝑦) ↔ ¬ 𝑦𝑅𝑥)) |
11 | 10 | ralbidva 3196 | . . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)) |
12 | 11 | rexbidva 3296 | . . 3 ⊢ (𝑅 Or 𝐴 → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑥𝑅𝑦) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)) |
13 | 12 | 3ad2ant1 1129 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑥𝑅𝑦) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)) |
14 | 1, 13 | mpbird 259 | 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 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ∅c0 4291 class class class wbr 5066 Or wor 5473 Fincfn 8509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-om 7581 df-1o 8102 df-er 8289 df-en 8510 df-fin 8513 |
This theorem is referenced by: fiinfg 8963 fiminre 11588 |
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