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Mirrors > Home > MPE Home > Th. List > cofcut1d | Structured version Visualization version GIF version |
Description: If 𝐶 is cofinal with 𝐴 and 𝐷 is coinitial with 𝐵 and the cut of 𝐴 and 𝐵 lies between 𝐶 and 𝐷, then the cut of 𝐶 and 𝐷 is equal to the cut of 𝐴 and 𝐵. Theorem 2.6 of [Gonshor] p. 10. (Contributed by Scott Fenton, 23-Jan-2025.) |
Ref | Expression |
---|---|
cofcut1d.1 | ⊢ (𝜑 → 𝐴 <<s 𝐵) |
cofcut1d.2 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦) |
cofcut1d.3 | ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) |
cofcut1d.4 | ⊢ (𝜑 → 𝐶 <<s {(𝐴 |s 𝐵)}) |
cofcut1d.5 | ⊢ (𝜑 → {(𝐴 |s 𝐵)} <<s 𝐷) |
Ref | Expression |
---|---|
cofcut1d | ⊢ (𝜑 → (𝐴 |s 𝐵) = (𝐶 |s 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cofcut1d.1 | . 2 ⊢ (𝜑 → 𝐴 <<s 𝐵) | |
2 | cofcut1d.2 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦) | |
3 | cofcut1d.3 | . 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) | |
4 | cofcut1d.4 | . 2 ⊢ (𝜑 → 𝐶 <<s {(𝐴 |s 𝐵)}) | |
5 | cofcut1d.5 | . 2 ⊢ (𝜑 → {(𝐴 |s 𝐵)} <<s 𝐷) | |
6 | cofcut1 27387 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷)) | |
7 | 1, 2, 3, 4, 5, 6 | syl122anc 1380 | 1 ⊢ (𝜑 → (𝐴 |s 𝐵) = (𝐶 |s 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∀wral 3062 ∃wrex 3071 {csn 4627 class class class wbr 5147 (class class class)co 7404 ≤s csle 27227 <<s csslt 27262 |s cscut 27264 |
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-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7720 |
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-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6364 df-on 6365 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1o 8461 df-2o 8462 df-no 27126 df-slt 27127 df-bday 27128 df-sle 27228 df-sslt 27263 df-scut 27265 |
This theorem is referenced by: addsuniflem 27464 negsunif 27509 mulsuniflem 27584 |
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