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Mirrors > Home > MPE Home > Th. List > cofcut2d | Structured version Visualization version GIF version |
Description: If 𝐴 and 𝐶 are mutually cofinal and 𝐵 and 𝐷 are mutually coinitial, then the cut of 𝐴 and 𝐵 is equal to the cut of 𝐶 and 𝐷. Theorem 2.7 of [Gonshor] p. 10. (Contributed by Scott Fenton, 23-Jan-2025.) |
Ref | Expression |
---|---|
cofcut2d.1 | ⊢ (𝜑 → 𝐴 <<s 𝐵) |
cofcut2d.2 | ⊢ (𝜑 → 𝐶 ∈ 𝒫 No ) |
cofcut2d.3 | ⊢ (𝜑 → 𝐷 ∈ 𝒫 No ) |
cofcut2d.4 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦) |
cofcut2d.5 | ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) |
cofcut2d.6 | ⊢ (𝜑 → ∀𝑡 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑡 ≤s 𝑢) |
cofcut2d.7 | ⊢ (𝜑 → ∀𝑟 ∈ 𝐷 ∃𝑠 ∈ 𝐵 𝑠 ≤s 𝑟) |
Ref | Expression |
---|---|
cofcut2d | ⊢ (𝜑 → (𝐴 |s 𝐵) = (𝐶 |s 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cofcut2d.1 | . 2 ⊢ (𝜑 → 𝐴 <<s 𝐵) | |
2 | cofcut2d.2 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝒫 No ) | |
3 | cofcut2d.3 | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝒫 No ) | |
4 | cofcut2d.4 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦) | |
5 | cofcut2d.5 | . 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) | |
6 | cofcut2d.6 | . 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑡 ≤s 𝑢) | |
7 | cofcut2d.7 | . 2 ⊢ (𝜑 → ∀𝑟 ∈ 𝐷 ∃𝑠 ∈ 𝐵 𝑠 ≤s 𝑟) | |
8 | cofcut2 27971 | . 2 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 ∈ 𝒫 No ∧ 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟 ∈ 𝐷 ∃𝑠 ∈ 𝐵 𝑠 ≤s 𝑟)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷)) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | syl322anc 1397 | 1 ⊢ (𝜑 → (𝐴 |s 𝐵) = (𝐶 |s 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 𝒫 cpw 4605 class class class wbr 5148 (class class class)co 7431 No csur 27699 ≤s csle 27804 <<s csslt 27840 |s cscut 27842 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1o 8505 df-2o 8506 df-no 27702 df-slt 27703 df-bday 27704 df-sle 27805 df-sslt 27841 df-scut 27843 |
This theorem is referenced by: cutlt 27981 |
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