Theorem ssltex1 33250

Description: The first argument of surreal set less than exists. (Contributed by Scott Fenton, 8-Dec-2021.)

Assertion

Ref	Expression
ssltex1	⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)

Proof of Theorem ssltex1

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brsslt 33249	. 2 ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)))
2		simpll 765	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦)) → 𝐴 ∈ V)
3	1, 2	sylbi 219	1 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   ∈ wcel 2110  ∀wral 3138  Vcvv 3494   ⊆ wss 3935   class class class wbr 5058   No csur 33142   <s cslt 33143   <<s csslt 33245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-xp 5555  df-sslt 33246

This theorem is referenced by:  sssslt1  33255  sssslt2  33256  conway  33259  scutval  33260  sslttr  33263  ssltun1  33264  ssltun2  33265  etasslt  33269  slerec  33272