Metamath Proof Explorer

Theorem onfrALTlem3

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

Ref Expression
Assertion onfrALTlem3 ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ( ( 𝑥 ∈ 𝑎 ∧ ¬ ( 𝑎 ∩ 𝑥 ) = ∅ ) → ∃ 𝑦 ∈ ( 𝑎 ∩ 𝑥 ) ( ( 𝑎 ∩ 𝑥 ) ∩ 𝑦 ) = ∅ ) )

Proof

Step Hyp Ref Expression
1 ssid ⊢ ( 𝑎 ∩ 𝑥 ) ⊆ ( 𝑎 ∩ 𝑥 )
2 simpr ⊢ ( ( 𝑥 ∈ 𝑎 ∧ ¬ ( 𝑎 ∩ 𝑥 ) = ∅ ) → ¬ ( 𝑎 ∩ 𝑥 ) = ∅ )
3 2 a1i ⊢ ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ( ( 𝑥 ∈ 𝑎 ∧ ¬ ( 𝑎 ∩ 𝑥 ) = ∅ ) → ¬ ( 𝑎 ∩ 𝑥 ) = ∅ ) )
4 df-ne ⊢ ( ( 𝑎 ∩ 𝑥 ) ≠ ∅ ↔ ¬ ( 𝑎 ∩ 𝑥 ) = ∅ )
5 3 4 syl6ibr ⊢ ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ( ( 𝑥 ∈ 𝑎 ∧ ¬ ( 𝑎 ∩ 𝑥 ) = ∅ ) → ( 𝑎 ∩ 𝑥 ) ≠ ∅ ) )
6 pm3.2 ⊢ ( ( 𝑎 ∩ 𝑥 ) ⊆ ( 𝑎 ∩ 𝑥 ) → ( ( 𝑎 ∩ 𝑥 ) ≠ ∅ → ( ( 𝑎 ∩ 𝑥 ) ⊆ ( 𝑎 ∩ 𝑥 ) ∧ ( 𝑎 ∩ 𝑥 ) ≠ ∅ ) ) )
7 1 5 6 mpsylsyld ⊢ ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ( ( 𝑥 ∈ 𝑎 ∧ ¬ ( 𝑎 ∩ 𝑥 ) = ∅ ) → ( ( 𝑎 ∩ 𝑥 ) ⊆ ( 𝑎 ∩ 𝑥 ) ∧ ( 𝑎 ∩ 𝑥 ) ≠ ∅ ) ) )
8 vex ⊢ 𝑥 ∈ V
9 8 inex2 ⊢ ( 𝑎 ∩ 𝑥 ) ∈ V
10 inss2 ⊢ ( 𝑎 ∩ 𝑥 ) ⊆ 𝑥
11 simpl ⊢ ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → 𝑎 ⊆ On )
12 simpl ⊢ ( ( 𝑥 ∈ 𝑎 ∧ ¬ ( 𝑎 ∩ 𝑥 ) = ∅ ) → 𝑥 ∈ 𝑎 )
13 ssel ⊢ ( 𝑎 ⊆ On → ( 𝑥 ∈ 𝑎 → 𝑥 ∈ On ) )
14 11 12 13 syl2im ⊢ ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ( ( 𝑥 ∈ 𝑎 ∧ ¬ ( 𝑎 ∩ 𝑥 ) = ∅ ) → 𝑥 ∈ On ) )
15 eloni ⊢ ( 𝑥 ∈ On → Ord 𝑥 )
16 14 15 syl6 ⊢ ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ( ( 𝑥 ∈ 𝑎 ∧ ¬ ( 𝑎 ∩ 𝑥 ) = ∅ ) → Ord 𝑥 ) )
17 ordwe ⊢ ( Ord 𝑥 → E We 𝑥 )
18 16 17 syl6 ⊢ ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ( ( 𝑥 ∈ 𝑎 ∧ ¬ ( 𝑎 ∩ 𝑥 ) = ∅ ) → E We 𝑥 ) )
19 wess ⊢ ( ( 𝑎 ∩ 𝑥 ) ⊆ 𝑥 → ( E We 𝑥 → E We ( 𝑎 ∩ 𝑥 ) ) )
20 10 18 19 mpsylsyld ⊢ ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ( ( 𝑥 ∈ 𝑎 ∧ ¬ ( 𝑎 ∩ 𝑥 ) = ∅ ) → E We ( 𝑎 ∩ 𝑥 ) ) )
21 wefr ⊢ ( E We ( 𝑎 ∩ 𝑥 ) → E Fr ( 𝑎 ∩ 𝑥 ) )
22 20 21 syl6 ⊢ ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ( ( 𝑥 ∈ 𝑎 ∧ ¬ ( 𝑎 ∩ 𝑥 ) = ∅ ) → E Fr ( 𝑎 ∩ 𝑥 ) ) )
23 dfepfr ⊢ ( E Fr ( 𝑎 ∩ 𝑥 ) ↔ ∀ 𝑏 ( ( 𝑏 ⊆ ( 𝑎 ∩ 𝑥 ) ∧ 𝑏 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑏 ( 𝑏 ∩ 𝑦 ) = ∅ ) )
24 22 23 syl6ib ⊢ ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ( ( 𝑥 ∈ 𝑎 ∧ ¬ ( 𝑎 ∩ 𝑥 ) = ∅ ) → ∀ 𝑏 ( ( 𝑏 ⊆ ( 𝑎 ∩ 𝑥 ) ∧ 𝑏 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑏 ( 𝑏 ∩ 𝑦 ) = ∅ ) ) )
25 spsbc ⊢ ( ( 𝑎 ∩ 𝑥 ) ∈ V → ( ∀ 𝑏 ( ( 𝑏 ⊆ ( 𝑎 ∩ 𝑥 ) ∧ 𝑏 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑏 ( 𝑏 ∩ 𝑦 ) = ∅ ) → [ ( 𝑎 ∩ 𝑥 ) / 𝑏 ] ( ( 𝑏 ⊆ ( 𝑎 ∩ 𝑥 ) ∧ 𝑏 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑏 ( 𝑏 ∩ 𝑦 ) = ∅ ) ) )
26 9 24 25 mpsylsyld ⊢ ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ( ( 𝑥 ∈ 𝑎 ∧ ¬ ( 𝑎 ∩ 𝑥 ) = ∅ ) → [ ( 𝑎 ∩ 𝑥 ) / 𝑏 ] ( ( 𝑏 ⊆ ( 𝑎 ∩ 𝑥 ) ∧ 𝑏 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑏 ( 𝑏 ∩ 𝑦 ) = ∅ ) ) )
27 onfrALTlem5 ⊢ ( [ ( 𝑎 ∩ 𝑥 ) / 𝑏 ] ( ( 𝑏 ⊆ ( 𝑎 ∩ 𝑥 ) ∧ 𝑏 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑏 ( 𝑏 ∩ 𝑦 ) = ∅ ) ↔ ( ( ( 𝑎 ∩ 𝑥 ) ⊆ ( 𝑎 ∩ 𝑥 ) ∧ ( 𝑎 ∩ 𝑥 ) ≠ ∅ ) → ∃ 𝑦 ∈ ( 𝑎 ∩ 𝑥 ) ( ( 𝑎 ∩ 𝑥 ) ∩ 𝑦 ) = ∅ ) )
28 26 27 syl6ib ⊢ ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ( ( 𝑥 ∈ 𝑎 ∧ ¬ ( 𝑎 ∩ 𝑥 ) = ∅ ) → ( ( ( 𝑎 ∩ 𝑥 ) ⊆ ( 𝑎 ∩ 𝑥 ) ∧ ( 𝑎 ∩ 𝑥 ) ≠ ∅ ) → ∃ 𝑦 ∈ ( 𝑎 ∩ 𝑥 ) ( ( 𝑎 ∩ 𝑥 ) ∩ 𝑦 ) = ∅ ) ) )
29 7 28 mpdd ⊢ ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ( ( 𝑥 ∈ 𝑎 ∧ ¬ ( 𝑎 ∩ 𝑥 ) = ∅ ) → ∃ 𝑦 ∈ ( 𝑎 ∩ 𝑥 ) ( ( 𝑎 ∩ 𝑥 ) ∩ 𝑦 ) = ∅ ) )