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Mirrors > Home > MPE Home > Th. List > Mathboxes > sssslt2 | Structured version Visualization version GIF version |
Description: Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.) |
Ref | Expression |
---|---|
sssslt2 | ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 <<s 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssltex1 33667 | . . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V) | |
2 | 1 | adantr 484 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 ∈ V) |
3 | ssltex2 33668 | . . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V) | |
4 | 3 | adantr 484 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐵 ∈ V) |
5 | simpr 488 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ 𝐵) | |
6 | 4, 5 | ssexd 5202 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ∈ V) |
7 | ssltss1 33669 | . . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No ) | |
8 | 7 | adantr 484 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 ⊆ No ) |
9 | ssltss2 33670 | . . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No ) | |
10 | 9 | adantr 484 | . . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐵 ⊆ No ) |
11 | 5, 10 | sstrd 3897 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ No ) |
12 | ssltsep 33671 | . . . 4 ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) | |
13 | ssralv 3953 | . . . . 5 ⊢ (𝐶 ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 → ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)) | |
14 | 13 | ralimdv 3091 | . . . 4 ⊢ (𝐶 ⊆ 𝐵 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)) |
15 | 12, 14 | mpan9 510 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦) |
16 | 8, 11, 15 | 3jca 1130 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐴 ⊆ No ∧ 𝐶 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)) |
17 | brsslt 33666 | . 2 ⊢ (𝐴 <<s 𝐶 ↔ ((𝐴 ∈ V ∧ 𝐶 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐶 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦))) | |
18 | 2, 6, 16, 17 | syl21anbrc 1346 | 1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 <<s 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2112 ∀wral 3051 Vcvv 3398 ⊆ wss 3853 class class class wbr 5039 No csur 33529 <s cslt 33530 <<s csslt 33661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-xp 5542 df-sslt 33662 |
This theorem is referenced by: scutun12 33690 |
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