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Theorem sltasym 31524
 Description: Surreal less than is asymmetric. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
sltasym ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 → ¬ 𝐵 <s 𝐴))

Proof of Theorem sltasym
StepHypRef Expression
1 sltirr 31522 . . . 4 (𝐴 No → ¬ 𝐴 <s 𝐴)
21ad2antrr 761 . . 3 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → ¬ 𝐴 <s 𝐴)
3 slttr 31523 . . . . 5 ((𝐴 No 𝐵 No 𝐴 No ) → ((𝐴 <s 𝐵𝐵 <s 𝐴) → 𝐴 <s 𝐴))
433anidm13 1381 . . . 4 ((𝐴 No 𝐵 No ) → ((𝐴 <s 𝐵𝐵 <s 𝐴) → 𝐴 <s 𝐴))
54expdimp 453 . . 3 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → (𝐵 <s 𝐴𝐴 <s 𝐴))
62, 5mtod 189 . 2 (((𝐴 No 𝐵 No ) ∧ 𝐴 <s 𝐵) → ¬ 𝐵 <s 𝐴)
76ex 450 1 ((𝐴 No 𝐵 No ) → (𝐴 <s 𝐵 → ¬ 𝐵 <s 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∈ wcel 1992   class class class wbr 4618   No csur 31486
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