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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 32808 | . 2 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))) | |
2 | simpr2 1186 | . 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) → 𝐵 ⊆ No ) | |
3 | 1, 2 | sylbi 218 | 1 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1078 ∈ wcel 2079 ∀wral 3103 Vcvv 3432 ⊆ wss 3854 class class class wbr 4956 No csur 32701 <s cslt 32702 <<s csslt 32804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pr 5214 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ral 3108 df-rex 3109 df-rab 3112 df-v 3434 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-sn 4467 df-pr 4469 df-op 4473 df-br 4957 df-opab 5019 df-xp 5441 df-sslt 32805 |
This theorem is referenced by: sssslt1 32814 sssslt2 32815 conway 32818 sslttr 32822 ssltun1 32823 ssltun2 32824 etasslt 32828 slerec 32831 sltrec 32832 |
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