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Mirrors > Home > MPE Home > Th. List > sssslt2 | Structured version Visualization version GIF version |
Description: Relation 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 27669 | . . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V) | |
2 | 1 | adantr 480 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 ∈ V) |
3 | ssltex2 27670 | . . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V) | |
4 | 3 | adantr 480 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐵 ∈ V) |
5 | simpr 484 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ 𝐵) | |
6 | 4, 5 | ssexd 5317 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ∈ V) |
7 | ssltss1 27671 | . . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No ) | |
8 | 7 | adantr 480 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 ⊆ No ) |
9 | ssltss2 27672 | . . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No ) | |
10 | 9 | adantr 480 | . . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐵 ⊆ No ) |
11 | 5, 10 | sstrd 3987 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ No ) |
12 | ssltsep 27673 | . . . 4 ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) | |
13 | ssralv 4045 | . . . . 5 ⊢ (𝐶 ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 → ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)) | |
14 | 13 | ralimdv 3163 | . . . 4 ⊢ (𝐶 ⊆ 𝐵 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)) |
15 | 12, 14 | mpan9 506 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦) |
16 | 8, 11, 15 | 3jca 1125 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐴 ⊆ No ∧ 𝐶 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)) |
17 | brsslt 27668 | . 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 1341 | 1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 <<s 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 ∈ wcel 2098 ∀wral 3055 Vcvv 3468 ⊆ wss 3943 class class class wbr 5141 No csur 27523 <s cslt 27524 <<s csslt 27663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-xp 5675 df-sslt 27664 |
This theorem is referenced by: scutun12 27693 |
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