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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 33151 | . 2 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))) | |
2 | simpr1 1186 | . 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) → 𝐴 ⊆ No ) | |
3 | 1, 2 | sylbi 218 | 1 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ⊆ wss 3933 class class class wbr 5057 No csur 33044 <s cslt 33045 <<s csslt 33147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-sslt 33148 |
This theorem is referenced by: sssslt1 33157 sssslt2 33158 conway 33161 scutval 33162 sslttr 33165 ssltun1 33166 ssltun2 33167 dmscut 33169 etasslt 33171 slerec 33174 sltrec 33175 |
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