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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 33276 | . . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V) | |
2 | 1 | adantr 483 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 ∈ V) |
3 | ssltex2 33277 | . . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V) | |
4 | 3 | adantr 483 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐵 ∈ V) |
5 | simpr 487 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ 𝐵) | |
6 | 4, 5 | ssexd 5221 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ∈ V) |
7 | ssltss1 33278 | . . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No ) | |
8 | 7 | adantr 483 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 ⊆ No ) |
9 | ssltss2 33279 | . . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No ) | |
10 | 9 | adantr 483 | . . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐵 ⊆ No ) |
11 | 5, 10 | sstrd 3970 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ No ) |
12 | ssltsep 33280 | . . . . 5 ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) | |
13 | 12 | adantr 483 | . . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) |
14 | ssralv 4026 | . . . . . 6 ⊢ (𝐶 ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 → ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)) | |
15 | 5, 14 | syl 17 | . . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → (∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 → ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)) |
16 | 15 | ralimdv 3177 | . . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)) |
17 | 13, 16 | mpd 15 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦) |
18 | 8, 11, 17 | 3jca 1123 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐴 ⊆ No ∧ 𝐶 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)) |
19 | brsslt 33275 | . 2 ⊢ (𝐴 <<s 𝐶 ↔ ((𝐴 ∈ V ∧ 𝐶 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐶 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦))) | |
20 | 2, 6, 18, 19 | syl21anbrc 1339 | 1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 <<s 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1082 ∈ wcel 2113 ∀wral 3137 Vcvv 3491 ⊆ wss 3929 class class class wbr 5059 No csur 33168 <s cslt 33169 <<s csslt 33271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-xp 5554 df-sslt 33272 |
This theorem is referenced by: scutun12 33292 |
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