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Mirrors > Home > MPE Home > Th. List > sssslt1 | Structured version Visualization version GIF version |
Description: Relation 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 27849 | . . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐴 ∈ V) |
3 | simpr 484 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ 𝐴) | |
4 | 2, 3 | ssexd 5342 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ∈ V) |
5 | ssltex2 27850 | . . 3 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V) | |
6 | 5 | adantr 480 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐵 ∈ V) |
7 | ssltss1 27851 | . . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No ) | |
8 | 7 | adantr 480 | . . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐴 ⊆ No ) |
9 | 3, 8 | sstrd 4019 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ No ) |
10 | ssltss2 27852 | . . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No ) | |
11 | 10 | adantr 480 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐵 ⊆ No ) |
12 | ssltsep 27853 | . . . 4 ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) | |
13 | ssralv 4077 | . . . 4 ⊢ (𝐶 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) | |
14 | 12, 13 | mpan9 506 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) |
15 | 9, 11, 14 | 3jca 1128 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐶 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) |
16 | brsslt 27848 | . 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 1344 | 1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 <<s 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 ∀wral 3067 Vcvv 3488 ⊆ wss 3976 class class class wbr 5166 No csur 27702 <s cslt 27703 <<s csslt 27843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-xp 5706 df-sslt 27844 |
This theorem is referenced by: scutun12 27873 cutmax 27986 precsexlem11 28259 |
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