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Theorem sleloe 32004
 Description: Surreal less than or equal in terms of less than. (Contributed by Scott Fenton, 8-Dec-2021.)
Assertion
Ref Expression
sleloe ((𝐴 No 𝐵 No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵𝐴 = 𝐵)))

Proof of Theorem sleloe
StepHypRef Expression
1 slenlt 32002 . 2 ((𝐴 No 𝐵 No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴))
2 orcom 401 . . . 4 ((𝐴 <s 𝐵𝐴 = 𝐵) ↔ (𝐴 = 𝐵𝐴 <s 𝐵))
3 eqcom 2658 . . . . 5 (𝐴 = 𝐵𝐵 = 𝐴)
43orbi1i 541 . . . 4 ((𝐴 = 𝐵𝐴 <s 𝐵) ↔ (𝐵 = 𝐴𝐴 <s 𝐵))
52, 4bitri 264 . . 3 ((𝐴 <s 𝐵𝐴 = 𝐵) ↔ (𝐵 = 𝐴𝐴 <s 𝐵))
6 sltso 31952 . . . . . 6 <s Or No
7 sotric 5090 . . . . . 6 (( <s Or No ∧ (𝐵 No 𝐴 No )) → (𝐵 <s 𝐴 ↔ ¬ (𝐵 = 𝐴𝐴 <s 𝐵)))
86, 7mpan 706 . . . . 5 ((𝐵 No 𝐴 No ) → (𝐵 <s 𝐴 ↔ ¬ (𝐵 = 𝐴𝐴 <s 𝐵)))
98ancoms 468 . . . 4 ((𝐴 No 𝐵 No ) → (𝐵 <s 𝐴 ↔ ¬ (𝐵 = 𝐴𝐴 <s 𝐵)))
109con2bid 343 . . 3 ((𝐴 No 𝐵 No ) → ((𝐵 = 𝐴𝐴 <s 𝐵) ↔ ¬ 𝐵 <s 𝐴))
115, 10syl5bb 272 . 2 ((𝐴 No 𝐵 No ) → ((𝐴 <s 𝐵𝐴 = 𝐵) ↔ ¬ 𝐵 <s 𝐴))
121, 11bitr4d 271 1 ((𝐴 No 𝐵 No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵𝐴 = 𝐵)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   = wceq 1523   ∈ wcel 2030   class class class wbr 4685   Or wor 5063   No csur 31918
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