| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > ssltex1 | Structured version Visualization version GIF version | ||
| Description: The first argument of surreal set less-than exists. (Contributed by Scott Fenton, 8-Dec-2021.) |
| Ref | Expression |
|---|---|
| ssltex1 | ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | brsslt 27830 | . 2 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))) | |
| 2 | simpll 767 | . 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) → 𝐴 ∈ V) | |
| 3 | 1, 2 | sylbi 217 | 1 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 No csur 27684 <s cslt 27685 <<s csslt 27825 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-sslt 27826 |
| This theorem is referenced by: sssslt1 27840 sssslt2 27841 conway 27844 scutval 27845 sslttr 27852 ssltun1 27853 ssltun2 27854 etasslt 27858 etasslt2 27859 scutbdaybnd2lim 27862 slerec 27864 madecut 27921 coinitsslt 27953 cofcut1 27954 cofcutr 27958 cutlt 27966 addsuniflem 28034 negsunif 28087 ssltmul1 28173 ssltmul2 28174 precsexlem11 28241 |
| Copyright terms: Public domain | W3C validator |