Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ssltss2 | Structured version Visualization version GIF version |
Description: The second argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.) |
Ref | Expression |
---|---|
ssltss2 | ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brsslt 33666 | . 2 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))) | |
2 | simpr2 1197 | . 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) → 𝐵 ⊆ No ) | |
3 | 1, 2 | sylbi 220 | 1 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2112 ∀wral 3051 Vcvv 3398 ⊆ wss 3853 class class class wbr 5039 No csur 33529 <s cslt 33530 <<s csslt 33661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-xp 5542 df-sslt 33662 |
This theorem is referenced by: sssslt1 33675 sssslt2 33676 conway 33679 sslttr 33687 ssltun1 33688 ssltun2 33689 etasslt 33693 slerec 33699 sltrec 33700 cofsslt 33774 coinitsslt 33775 cofcut1 33776 cofcutr 33778 |
Copyright terms: Public domain | W3C validator |