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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 32816 | . . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V) | |
2 | 1 | adantr 473 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 ∈ V) |
3 | ssltex2 32817 | . . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V) | |
4 | 3 | adantr 473 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐵 ∈ V) |
5 | simpr 477 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ 𝐵) | |
6 | 4, 5 | ssexd 5088 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ∈ V) |
7 | ssltss1 32818 | . . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No ) | |
8 | 7 | adantr 473 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 ⊆ No ) |
9 | ssltss2 32819 | . . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No ) | |
10 | 9 | adantr 473 | . . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐵 ⊆ No ) |
11 | 5, 10 | sstrd 3870 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ No ) |
12 | ssltsep 32820 | . . . . 5 ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) | |
13 | 12 | adantr 473 | . . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) |
14 | ssralv 3925 | . . . . . 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 3130 | . . . 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 1109 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐴 ⊆ No ∧ 𝐶 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 𝑥 <s 𝑦)) |
19 | brsslt 32815 | . 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 1325 | 1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 <<s 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1069 ∈ wcel 2051 ∀wral 3090 Vcvv 3417 ⊆ wss 3831 class class class wbr 4934 No csur 32708 <s cslt 32709 <<s csslt 32811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-sep 5064 ax-nul 5071 ax-pr 5190 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3419 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4182 df-if 4354 df-sn 4445 df-pr 4447 df-op 4451 df-br 4935 df-opab 4997 df-xp 5417 df-sslt 32812 |
This theorem is referenced by: scutun12 32832 |
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