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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 27831 | . . . 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 5324 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ∈ V) |
| 5 | ssltex2 27832 | . . 3 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V) | |
| 6 | 5 | adantr 480 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐵 ∈ V) |
| 7 | ssltss1 27833 | . . . . 5 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No ) | |
| 8 | 7 | adantr 480 | . . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐴 ⊆ No ) |
| 9 | 3, 8 | sstrd 3994 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ No ) |
| 10 | ssltss2 27834 | . . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No ) | |
| 11 | 10 | adantr 480 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐵 ⊆ No ) |
| 12 | ssltsep 27835 | . . . 4 ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) | |
| 13 | ssralv 4052 | . . . 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 1129 | . 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐶 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) |
| 16 | brsslt 27830 | . 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 1345 | 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 3061 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 No csur 27684 <s cslt 27685 <<s csslt 27825 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-sslt 27826 |
| This theorem is referenced by: scutun12 27855 cutmax 27968 precsexlem11 28241 |
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