Theorem nosepeq

Description: The values of two surreals at a point less than their separators are equal. (Contributed by Scott Fenton, 6-Dec-2021)

	Ref	Expression
Assertion	nosepeq	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵 ) ∧ 𝑋 ∈ ∩ { 𝑥 ∈ On ∣ ( 𝐴 ‘ 𝑥 ) ≠ ( 𝐵 ‘ 𝑥 ) } ) → ( 𝐴 ‘ 𝑋 ) = ( 𝐵 ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		nosepon	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵 ) → ∩ { 𝑥 ∈ On ∣ ( 𝐴 ‘ 𝑥 ) ≠ ( 𝐵 ‘ 𝑥 ) } ∈ On )
2		onelon	⊢ ( ( ∩ { 𝑥 ∈ On ∣ ( 𝐴 ‘ 𝑥 ) ≠ ( 𝐵 ‘ 𝑥 ) } ∈ On ∧ 𝑋 ∈ ∩ { 𝑥 ∈ On ∣ ( 𝐴 ‘ 𝑥 ) ≠ ( 𝐵 ‘ 𝑥 ) } ) → 𝑋 ∈ On )
3	1 2	sylan	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵 ) ∧ 𝑋 ∈ ∩ { 𝑥 ∈ On ∣ ( 𝐴 ‘ 𝑥 ) ≠ ( 𝐵 ‘ 𝑥 ) } ) → 𝑋 ∈ On )
4		simpr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵 ) ∧ 𝑋 ∈ ∩ { 𝑥 ∈ On ∣ ( 𝐴 ‘ 𝑥 ) ≠ ( 𝐵 ‘ 𝑥 ) } ) → 𝑋 ∈ ∩ { 𝑥 ∈ On ∣ ( 𝐴 ‘ 𝑥 ) ≠ ( 𝐵 ‘ 𝑥 ) } )
5		fveq2	⊢ ( 𝑥 = 𝑋 → ( 𝐴 ‘ 𝑥 ) = ( 𝐴 ‘ 𝑋 ) )
6		fveq2	⊢ ( 𝑥 = 𝑋 → ( 𝐵 ‘ 𝑥 ) = ( 𝐵 ‘ 𝑋 ) )
7	5 6	neeq12d	⊢ ( 𝑥 = 𝑋 → ( ( 𝐴 ‘ 𝑥 ) ≠ ( 𝐵 ‘ 𝑥 ) ↔ ( 𝐴 ‘ 𝑋 ) ≠ ( 𝐵 ‘ 𝑋 ) ) )
8	7	onnminsb	⊢ ( 𝑋 ∈ On → ( 𝑋 ∈ ∩ { 𝑥 ∈ On ∣ ( 𝐴 ‘ 𝑥 ) ≠ ( 𝐵 ‘ 𝑥 ) } → ¬ ( 𝐴 ‘ 𝑋 ) ≠ ( 𝐵 ‘ 𝑋 ) ) )
9	3 4 8	sylc	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵 ) ∧ 𝑋 ∈ ∩ { 𝑥 ∈ On ∣ ( 𝐴 ‘ 𝑥 ) ≠ ( 𝐵 ‘ 𝑥 ) } ) → ¬ ( 𝐴 ‘ 𝑋 ) ≠ ( 𝐵 ‘ 𝑋 ) )
10		df-ne	⊢ ( ( 𝐴 ‘ 𝑋 ) ≠ ( 𝐵 ‘ 𝑋 ) ↔ ¬ ( 𝐴 ‘ 𝑋 ) = ( 𝐵 ‘ 𝑋 ) )
11	10	con2bii	⊢ ( ( 𝐴 ‘ 𝑋 ) = ( 𝐵 ‘ 𝑋 ) ↔ ¬ ( 𝐴 ‘ 𝑋 ) ≠ ( 𝐵 ‘ 𝑋 ) )
12	9 11	sylibr	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵 ) ∧ 𝑋 ∈ ∩ { 𝑥 ∈ On ∣ ( 𝐴 ‘ 𝑥 ) ≠ ( 𝐵 ‘ 𝑥 ) } ) → ( 𝐴 ‘ 𝑋 ) = ( 𝐵 ‘ 𝑋 ) )