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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssltd | Structured version Visualization version GIF version |
Description: Deduce surreal set less than. (Contributed by Scott Fenton, 24-Sep-2024.) |
Ref | Expression |
---|---|
ssltd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
ssltd.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
ssltd.3 | ⊢ (𝜑 → 𝐴 ⊆ No ) |
ssltd.4 | ⊢ (𝜑 → 𝐵 ⊆ No ) |
ssltd.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 <s 𝑦) |
Ref | Expression |
---|---|
ssltd | ⊢ (𝜑 → 𝐴 <<s 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssltd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | 1 | elexd 3420 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
3 | ssltd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
4 | 3 | elexd 3420 | . 2 ⊢ (𝜑 → 𝐵 ∈ V) |
5 | ssltd.3 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ No ) | |
6 | ssltd.4 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ No ) | |
7 | ssltd.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 <s 𝑦) | |
8 | 7 | 3expb 1121 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 <s 𝑦) |
9 | 8 | ralrimivva 3104 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) |
10 | 5, 6, 9 | 3jca 1129 | . 2 ⊢ (𝜑 → (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) |
11 | brsslt 33635 | . 2 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))) | |
12 | 2, 4, 10, 11 | syl21anbrc 1345 | 1 ⊢ (𝜑 → 𝐴 <<s 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 ∈ wcel 2114 ∀wral 3054 Vcvv 3400 ⊆ wss 3853 class class class wbr 5040 No csur 33498 <s cslt 33499 <<s csslt 33630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-ral 3059 df-rex 3060 df-v 3402 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-br 5041 df-opab 5103 df-xp 5541 df-sslt 33631 |
This theorem is referenced by: nulsslt 33646 nulssgt 33647 sslttr 33656 ssltun1 33657 ssltun2 33658 ssltleft 33709 ssltright 33710 cofsslt 33743 coinitsslt 33744 cofcutr 33747 |
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