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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 3452 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
3 | ssltd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
4 | 3 | elexd 3452 | . 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 1119 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 <s 𝑦) |
9 | 8 | ralrimivva 3123 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) |
10 | 5, 6, 9 | 3jca 1127 | . 2 ⊢ (𝜑 → (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) |
11 | brsslt 33980 | . 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 1343 | 1 ⊢ (𝜑 → 𝐴 <<s 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 ∈ wcel 2106 ∀wral 3064 Vcvv 3432 ⊆ wss 3887 class class class wbr 5074 No csur 33843 <s cslt 33844 <<s csslt 33975 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-xp 5595 df-sslt 33976 |
This theorem is referenced by: nulsslt 33991 nulssgt 33992 sslttr 34001 ssltun1 34002 ssltun2 34003 ssltleft 34054 ssltright 34055 cofsslt 34088 coinitsslt 34089 cofcutr 34092 |
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