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Mirrors > Home > MPE Home > Th. List > fiinf2g | Structured version Visualization version GIF version |
Description: A finite set satisfies the conditions to have an infimum. (Contributed by AV, 6-Oct-2020.) |
Ref | Expression |
---|---|
fiinf2g | ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | soss 5601 | . . . . 5 ⊢ (𝐵 ⊆ 𝐴 → (𝑅 Or 𝐴 → 𝑅 Or 𝐵)) | |
2 | simp1 1136 | . . . . . . 7 ⊢ ((𝑅 Or 𝐵 ∧ 𝐵 ∈ Fin ∧ 𝐵 ≠ ∅) → 𝑅 Or 𝐵) | |
3 | fiinfg 9476 | . . . . . . 7 ⊢ ((𝑅 Or 𝐵 ∧ 𝐵 ∈ Fin ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) | |
4 | 2, 3 | infeu 9473 | . . . . . 6 ⊢ ((𝑅 Or 𝐵 ∧ 𝐵 ∈ Fin ∧ 𝐵 ≠ ∅) → ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) |
5 | 4 | 3exp 1119 | . . . . 5 ⊢ (𝑅 Or 𝐵 → (𝐵 ∈ Fin → (𝐵 ≠ ∅ → ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))))) |
6 | 1, 5 | syl6 35 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝑅 Or 𝐴 → (𝐵 ∈ Fin → (𝐵 ≠ ∅ → ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))))) |
7 | 6 | com4l 92 | . . 3 ⊢ (𝑅 Or 𝐴 → (𝐵 ∈ Fin → (𝐵 ≠ ∅ → (𝐵 ⊆ 𝐴 → ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))))) |
8 | 7 | 3imp2 1349 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) |
9 | reurex 3379 | . 2 ⊢ (∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) | |
10 | breq1 5144 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧𝑅𝑦 ↔ 𝑥𝑅𝑦)) | |
11 | 10 | rspcev 3609 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥𝑅𝑦) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) |
12 | 11 | ex 413 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 → (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) |
13 | 12 | ralrimivw 3149 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) |
14 | 13 | a1d 25 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) → ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) |
15 | 14 | anim2d 612 | . . 3 ⊢ (𝑥 ∈ 𝐵 → ((∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) → (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))) |
16 | 15 | reximia 3080 | . 2 ⊢ (∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) |
17 | 8, 9, 16 | 3syl 18 | 1 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1087 ∈ wcel 2106 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 ∃!wreu 3373 ⊆ wss 3944 ∅c0 4318 class class class wbr 5141 Or wor 5580 Fincfn 8922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-om 7839 df-en 8923 df-fin 8926 |
This theorem is referenced by: ballotlemsup 33334 |
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