Theorem onfrALTlem1

Description: Lemma for onfrALT . (Contributed by Alan Sare, 22-Jul-2012) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	onfrALTlem1	⊢ ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ( ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ) )

Proof

Step	Hyp	Ref	Expression
1		19.8a	⊢ ( ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) → ∃ 𝑥 ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) )
2	1	a1i	⊢ ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ( ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) → ∃ 𝑥 ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ) )
3		cbvexsv	⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ ∃ 𝑦 [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) )
4	2 3	imbitrdi	⊢ ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ( ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) → ∃ 𝑦 [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ) )
5		sbsbc	⊢ ( [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) )
6		onfrALTlem4	⊢ ( [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ ( 𝑦 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑦 ) = ∅ ) )
7	5 6	bitri	⊢ ( [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ ( 𝑦 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑦 ) = ∅ ) )
8	7	exbii	⊢ ( ∃ 𝑦 [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ ∃ 𝑦 ( 𝑦 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑦 ) = ∅ ) )
9	4 8	imbitrdi	⊢ ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ( ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) → ∃ 𝑦 ( 𝑦 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑦 ) = ∅ ) ) )
10		df-rex	⊢ ( ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ↔ ∃ 𝑦 ( 𝑦 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑦 ) = ∅ ) )
11	9 10	imbitrrdi	⊢ ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ( ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ) )