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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssltex2 | Structured version Visualization version GIF version |
Description: The second argument of surreal set less than exists. (Contributed by Scott Fenton, 8-Dec-2021.) |
Ref | Expression |
---|---|
ssltex2 | ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brsslt 32870 | . 2 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))) | |
2 | simplr 765 | . 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) → 𝐵 ∈ V) | |
3 | 1, 2 | sylbi 218 | 1 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1080 ∈ wcel 2081 ∀wral 3105 Vcvv 3437 ⊆ wss 3863 class class class wbr 4966 No csur 32763 <s cslt 32764 <<s csslt 32866 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5099 ax-nul 5106 ax-pr 5226 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-nul 4216 df-if 4386 df-sn 4477 df-pr 4479 df-op 4483 df-br 4967 df-opab 5029 df-xp 5454 df-sslt 32867 |
This theorem is referenced by: sssslt1 32876 sssslt2 32877 conway 32880 scutval 32881 sslttr 32884 ssltun1 32885 ssltun2 32886 etasslt 32890 slerec 32893 |
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