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Mirrors > Home > MPE Home > Th. List > Mathboxes > sssslt1 | Structured version Visualization version GIF version |
Description: Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.) |
Ref | Expression |
---|---|
sssslt1 | ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 <<s 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssltex1 33718 | . . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V) | |
2 | 1 | adantr 484 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐴 ∈ V) |
3 | simpr 488 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ 𝐴) | |
4 | 2, 3 | ssexd 5217 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ∈ V) |
5 | ssltex2 33719 | . . 3 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V) | |
6 | 5 | adantr 484 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐵 ∈ V) |
7 | ssltss1 33720 | . . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No ) | |
8 | 7 | adantr 484 | . . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐴 ⊆ No ) |
9 | 3, 8 | sstrd 3911 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ No ) |
10 | ssltss2 33721 | . . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No ) | |
11 | 10 | adantr 484 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐵 ⊆ No ) |
12 | ssltsep 33722 | . . . 4 ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) | |
13 | ssralv 3967 | . . . 4 ⊢ (𝐶 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) | |
14 | 12, 13 | mpan9 510 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) |
15 | 9, 11, 14 | 3jca 1130 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐶 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) |
16 | brsslt 33717 | . 2 ⊢ (𝐶 <<s 𝐵 ↔ ((𝐶 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))) | |
17 | 4, 6, 15, 16 | syl21anbrc 1346 | 1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 <<s 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2110 ∀wral 3061 Vcvv 3408 ⊆ wss 3866 class class class wbr 5053 No csur 33580 <s cslt 33581 <<s csslt 33712 |
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 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
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 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-xp 5557 df-sslt 33713 |
This theorem is referenced by: scutun12 33741 |
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