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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssltss1 | Structured version Visualization version GIF version |
Description: The first argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.) |
Ref | Expression |
---|---|
ssltss1 | ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brsslt 33258 | . 2 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))) | |
2 | simpr1 1190 | . 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) → 𝐴 ⊆ No ) | |
3 | 1, 2 | sylbi 219 | 1 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2113 ∀wral 3141 Vcvv 3497 ⊆ wss 3939 class class class wbr 5069 No csur 33151 <s cslt 33152 <<s csslt 33254 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-xp 5564 df-sslt 33255 |
This theorem is referenced by: sssslt1 33264 sssslt2 33265 conway 33268 scutval 33269 sslttr 33272 ssltun1 33273 ssltun2 33274 dmscut 33276 etasslt 33278 slerec 33281 sltrec 33282 |
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