Theorem pexmidlem5N

Description: Lemma for pexmidN . (Contributed by NM, 2-Feb-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pexmidlem.l	⊢ ≤ = ( le ‘ 𝐾 )
	pexmidlem.j	⊢ ∨ = ( join ‘ 𝐾 )
	pexmidlem.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	pexmidlem.p	⊢ + = ( +𝑃 ‘ 𝐾 )
	pexmidlem.o	⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
	pexmidlem.m	⊢ 𝑀 = ( 𝑋 + { 𝑝 } )
Assertion	pexmidlem5N	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴 ) ∧ ( 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) ) ) → ( ( ⊥ ‘ 𝑋 ) ∩ 𝑀 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		pexmidlem.l	⊢ ≤ = ( le ‘ 𝐾 )
2		pexmidlem.j	⊢ ∨ = ( join ‘ 𝐾 )
3		pexmidlem.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4		pexmidlem.p	⊢ + = ( +𝑃 ‘ 𝐾 )
5		pexmidlem.o	⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
6		pexmidlem.m	⊢ 𝑀 = ( 𝑋 + { 𝑝 } )
7		n0	⊢ ( ( ( ⊥ ‘ 𝑋 ) ∩ 𝑀 ) ≠ ∅ ↔ ∃ 𝑞 𝑞 ∈ ( ( ⊥ ‘ 𝑋 ) ∩ 𝑀 ) )
8	1 2 3 4 5 6	pexmidlem4N	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴 ) ∧ ( 𝑋 ≠ ∅ ∧ 𝑞 ∈ ( ( ⊥ ‘ 𝑋 ) ∩ 𝑀 ) ) ) → 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) )
9	8	expr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴 ) ∧ 𝑋 ≠ ∅ ) → ( 𝑞 ∈ ( ( ⊥ ‘ 𝑋 ) ∩ 𝑀 ) → 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) ) )
10	9	exlimdv	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴 ) ∧ 𝑋 ≠ ∅ ) → ( ∃ 𝑞 𝑞 ∈ ( ( ⊥ ‘ 𝑋 ) ∩ 𝑀 ) → 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) ) )
11	7 10	syl5bi	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴 ) ∧ 𝑋 ≠ ∅ ) → ( ( ( ⊥ ‘ 𝑋 ) ∩ 𝑀 ) ≠ ∅ → 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) ) )
12	11	necon1bd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴 ) ∧ 𝑋 ≠ ∅ ) → ( ¬ 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) → ( ( ⊥ ‘ 𝑋 ) ∩ 𝑀 ) = ∅ ) )
13	12	impr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴 ) ∧ ( 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ ( 𝑋 + ( ⊥ ‘ 𝑋 ) ) ) ) → ( ( ⊥ ‘ 𝑋 ) ∩ 𝑀 ) = ∅ )