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Mirrors > Home > MPE Home > Th. List > cofss | Structured version Visualization version GIF version |
Description: Cofinality for a subset. (Contributed by Scott Fenton, 13-Mar-2025.) |
Ref | Expression |
---|---|
cofss.1 | ⊢ (𝜑 → 𝐴 ⊆ No ) |
cofss.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
Ref | Expression |
---|---|
cofss | ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 ≤s 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cofss.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
2 | 1 | sselda 3995 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐴) |
3 | cofss.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ No ) | |
4 | 1, 3 | sstrd 4006 | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ No ) |
5 | 4 | sselda 3995 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ No ) |
6 | slerflex 27823 | . . . . 5 ⊢ (𝑧 ∈ No → 𝑧 ≤s 𝑧) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ≤s 𝑧) |
8 | breq2 5152 | . . . . 5 ⊢ (𝑦 = 𝑧 → (𝑧 ≤s 𝑦 ↔ 𝑧 ≤s 𝑧)) | |
9 | 8 | rspcev 3622 | . . . 4 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑧 ≤s 𝑧) → ∃𝑦 ∈ 𝐴 𝑧 ≤s 𝑦) |
10 | 2, 7, 9 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ∃𝑦 ∈ 𝐴 𝑧 ≤s 𝑦) |
11 | 10 | ralrimiva 3144 | . 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑧 ≤s 𝑦) |
12 | breq1 5151 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ≤s 𝑦 ↔ 𝑧 ≤s 𝑦)) | |
13 | 12 | rexbidv 3177 | . . 3 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐴 𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑧 ≤s 𝑦)) |
14 | 13 | cbvralvw 3235 | . 2 ⊢ (∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 ≤s 𝑦 ↔ ∀𝑧 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑧 ≤s 𝑦) |
15 | 11, 14 | sylibr 234 | 1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 ≤s 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 class class class wbr 5148 No csur 27699 ≤s csle 27804 |
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-sep 5302 ax-nul 5312 ax-pr 5438 |
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-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-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-fv 6571 df-1o 8505 df-2o 8506 df-no 27702 df-slt 27703 df-sle 27805 |
This theorem is referenced by: cutlt 27981 |
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