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Mirrors > Home > MPE Home > Th. List > cutmax | Structured version Visualization version GIF version |
Description: If 𝐴 has a maximum, then the maximum may be used alone in the cut. (Contributed by Scott Fenton, 20-Aug-2025.) |
Ref | Expression |
---|---|
cutmax.1 | ⊢ (𝜑 → 𝐴 <<s 𝐵) |
cutmax.2 | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
cutmax.3 | ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑋) |
Ref | Expression |
---|---|
cutmax | ⊢ (𝜑 → (𝐴 |s 𝐵) = ({𝑋} |s 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cutmax.1 | . 2 ⊢ (𝜑 → 𝐴 <<s 𝐵) | |
2 | cutmax.3 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑋) | |
3 | cutmax.2 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
4 | breq2 5170 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑦 ≤s 𝑥 ↔ 𝑦 ≤s 𝑋)) | |
5 | 4 | rexsng 4698 | . . . . 5 ⊢ (𝑋 ∈ 𝐴 → (∃𝑥 ∈ {𝑋}𝑦 ≤s 𝑥 ↔ 𝑦 ≤s 𝑋)) |
6 | 3, 5 | syl 17 | . . . 4 ⊢ (𝜑 → (∃𝑥 ∈ {𝑋}𝑦 ≤s 𝑥 ↔ 𝑦 ≤s 𝑋)) |
7 | 6 | ralbidv 3184 | . . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 ∃𝑥 ∈ {𝑋}𝑦 ≤s 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑋)) |
8 | 2, 7 | mpbird 257 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ∃𝑥 ∈ {𝑋}𝑦 ≤s 𝑥) |
9 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
10 | ssltss2 27854 | . . . . . . 7 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No ) | |
11 | 1, 10 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ No ) |
12 | 11 | sselda 4008 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ No ) |
13 | slerflex 27828 | . . . . 5 ⊢ (𝑥 ∈ No → 𝑥 ≤s 𝑥) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ≤s 𝑥) |
15 | breq1 5169 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ≤s 𝑥 ↔ 𝑥 ≤s 𝑥)) | |
16 | 15 | rspcev 3635 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ≤s 𝑥) → ∃𝑦 ∈ 𝐵 𝑦 ≤s 𝑥) |
17 | 9, 14, 16 | syl2anc 583 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 𝑦 ≤s 𝑥) |
18 | 17 | ralrimiva 3152 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 𝑦 ≤s 𝑥) |
19 | scutcut 27866 | . . . . 5 ⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵)) | |
20 | 1, 19 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵)) |
21 | 20 | simp2d 1143 | . . 3 ⊢ (𝜑 → 𝐴 <<s {(𝐴 |s 𝐵)}) |
22 | 3 | snssd 4834 | . . 3 ⊢ (𝜑 → {𝑋} ⊆ 𝐴) |
23 | sssslt1 27860 | . . 3 ⊢ ((𝐴 <<s {(𝐴 |s 𝐵)} ∧ {𝑋} ⊆ 𝐴) → {𝑋} <<s {(𝐴 |s 𝐵)}) | |
24 | 21, 22, 23 | syl2anc 583 | . 2 ⊢ (𝜑 → {𝑋} <<s {(𝐴 |s 𝐵)}) |
25 | 20 | simp3d 1144 | . 2 ⊢ (𝜑 → {(𝐴 |s 𝐵)} <<s 𝐵) |
26 | 1, 8, 18, 24, 25 | cofcut1d 27975 | 1 ⊢ (𝜑 → (𝐴 |s 𝐵) = ({𝑋} |s 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ∃wrex 3076 ⊆ wss 3976 {csn 4648 class class class wbr 5166 (class class class)co 7450 No csur 27704 ≤s csle 27809 <<s csslt 27845 |s cscut 27847 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-ord 6400 df-on 6401 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-1o 8524 df-2o 8525 df-no 27707 df-slt 27708 df-bday 27709 df-sle 27810 df-sslt 27846 df-scut 27848 |
This theorem is referenced by: (None) |
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