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Mirrors > Home > MPE Home > Th. List > 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 27845 | . 2 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))) | |
2 | simpr2 1194 | . 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) → 𝐵 ⊆ No ) | |
3 | 1, 2 | sylbi 217 | 1 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ⊆ wss 3963 class class class wbr 5148 No csur 27699 <s cslt 27700 <<s csslt 27840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-sslt 27841 |
This theorem is referenced by: sssslt1 27855 sssslt2 27856 conway 27859 sslttr 27867 ssltun1 27868 ssltun2 27869 etasslt 27873 slerec 27879 sltrec 27880 cofsslt 27967 coinitsslt 27968 cofcut1 27969 cofcutr 27973 cutlt 27981 cutmax 27983 addsuniflem 28049 negsunif 28102 ssltmul1 28188 ssltmul2 28189 mulsuniflem 28190 mulsunif2lem 28210 precsexlem11 28256 renegscl 28445 |
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