Metamath Proof Explorer

Theorem noinfbnd1lem4

Description: Lemma for noinfbnd1 . If U is a prolongment of T and in B , then ( Udom T ) is not undefined. (Contributed by Scott Fenton, 9-Aug-2024)

Ref Expression
Hypothesis noinfbnd1.1 ⊢ 𝑇 = if ( ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 , ( ( ℩ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ) ∪ { ⟨ dom ( ℩ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ) , 1o ⟩ } ) , ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐵 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) )
Assertion noinfbnd1lem4 ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) → ( 𝑈 ‘ dom 𝑇 ) ≠ ∅ )

Proof

Step Hyp Ref Expression
1 noinfbnd1.1 ⊢ 𝑇 = if ( ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 , ( ( ℩ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ) ∪ { ⟨ dom ( ℩ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ) , 1o ⟩ } ) , ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐵 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) )
2 simpl1 ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 )
3 simpl2 ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) )
4 simprl ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → 𝑤 ∈ 𝐵 )
5 simpl3 ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) )
6 simp2l ⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) → 𝐵 ⊆ No )
7 6 sselda ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ 𝑤 ∈ 𝐵 ) → 𝑤 ∈ No )
8 simp3l ⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) → 𝑈 ∈ 𝐵 )
9 6 8 sseldd ⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) → 𝑈 ∈ No )
10 9 adantr ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ 𝑤 ∈ 𝐵 ) → 𝑈 ∈ No )
11 ltsso ⊢ <s Or No
12 soasym ⊢ ( ( <s Or No ∧ ( 𝑤 ∈ No ∧ 𝑈 ∈ No ) ) → ( 𝑤 <s 𝑈 → ¬ 𝑈 <s 𝑤 ) )
13 11 12 mpan ⊢ ( ( 𝑤 ∈ No ∧ 𝑈 ∈ No ) → ( 𝑤 <s 𝑈 → ¬ 𝑈 <s 𝑤 ) )
14 7 10 13 syl2anc ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ 𝑤 ∈ 𝐵 ) → ( 𝑤 <s 𝑈 → ¬ 𝑈 <s 𝑤 ) )
15 14 impr ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ¬ 𝑈 <s 𝑤 )
16 4 15 jca ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑤 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑤 ) )
17 1 noinfbnd1lem2 ⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ∧ ( 𝑤 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑤 ) ) ) → ( 𝑤 ↾ dom 𝑇 ) = 𝑇 )
18 2 3 5 16 17 syl112anc ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑤 ↾ dom 𝑇 ) = 𝑇 )
19 1 noinfbnd1lem3 ⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑤 ∈ 𝐵 ∧ ( 𝑤 ↾ dom 𝑇 ) = 𝑇 ) ) → ( 𝑤 ‘ dom 𝑇 ) ≠ 1o )
20 2 3 4 18 19 syl112anc ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑤 ‘ dom 𝑇 ) ≠ 1o )
21 20 neneqd ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ¬ ( 𝑤 ‘ dom 𝑇 ) = 1o )
22 21 expr ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ 𝑤 ∈ 𝐵 ) → ( 𝑤 <s 𝑈 → ¬ ( 𝑤 ‘ dom 𝑇 ) = 1o ) )
23 imnan ⊢ ( ( 𝑤 <s 𝑈 → ¬ ( 𝑤 ‘ dom 𝑇 ) = 1o ) ↔ ¬ ( 𝑤 <s 𝑈 ∧ ( 𝑤 ‘ dom 𝑇 ) = 1o ) )
24 22 23 sylib ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ 𝑤 ∈ 𝐵 ) → ¬ ( 𝑤 <s 𝑈 ∧ ( 𝑤 ‘ dom 𝑇 ) = 1o ) )
25 24 nrexdv ⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) → ¬ ∃ 𝑤 ∈ 𝐵 ( 𝑤 <s 𝑈 ∧ ( 𝑤 ‘ dom 𝑇 ) = 1o ) )
26 breq2 ⊢ ( 𝑥 = 𝑈 → ( 𝑦 <s 𝑥 ↔ 𝑦 <s 𝑈 ) )
27 26 rexbidv ⊢ ( 𝑥 = 𝑈 → ( ∃ 𝑦 ∈ 𝐵 𝑦 <s 𝑥 ↔ ∃ 𝑦 ∈ 𝐵 𝑦 <s 𝑈 ) )
28 simpl1 ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 )
29 dfral2 ⊢ ( ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 𝑦 <s 𝑥 ↔ ¬ ∃ 𝑥 ∈ 𝐵 ¬ ∃ 𝑦 ∈ 𝐵 𝑦 <s 𝑥 )
30 ralnex ⊢ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ↔ ¬ ∃ 𝑦 ∈ 𝐵 𝑦 <s 𝑥 )
31 30 rexbii ⊢ ( ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ↔ ∃ 𝑥 ∈ 𝐵 ¬ ∃ 𝑦 ∈ 𝐵 𝑦 <s 𝑥 )
32 29 31 xchbinxr ⊢ ( ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 𝑦 <s 𝑥 ↔ ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 )
33 28 32 sylibr ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 𝑦 <s 𝑥 )
34 simpl3l ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → 𝑈 ∈ 𝐵 )
35 27 33 34 rspcdva ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → ∃ 𝑦 ∈ 𝐵 𝑦 <s 𝑈 )
36 breq1 ⊢ ( 𝑦 = 𝑤 → ( 𝑦 <s 𝑈 ↔ 𝑤 <s 𝑈 ) )
37 36 cbvrexvw ⊢ ( ∃ 𝑦 ∈ 𝐵 𝑦 <s 𝑈 ↔ ∃ 𝑤 ∈ 𝐵 𝑤 <s 𝑈 )
38 35 37 sylib ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → ∃ 𝑤 ∈ 𝐵 𝑤 <s 𝑈 )
39 simpl2l ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → 𝐵 ⊆ No )
40 39 adantr ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → 𝐵 ⊆ No )
41 simprl ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → 𝑤 ∈ 𝐵 )
42 40 41 sseldd ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → 𝑤 ∈ No )
43 34 adantr ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → 𝑈 ∈ 𝐵 )
44 40 43 sseldd ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → 𝑈 ∈ No )
45 simpl2 ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) )
46 1 noinfno ⊢ ( ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) → 𝑇 ∈ No )
47 45 46 syl ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → 𝑇 ∈ No )
48 47 adantr ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → 𝑇 ∈ No )
49 nodmon ⊢ ( 𝑇 ∈ No → dom 𝑇 ∈ On )
50 48 49 syl ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → dom 𝑇 ∈ On )
51 simpll1 ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 )
52 simpll2 ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) )
53 simpll3 ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) )
54 simprr ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → 𝑤 <s 𝑈 )
55 42 44 13 syl2anc ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑤 <s 𝑈 → ¬ 𝑈 <s 𝑤 ) )
56 54 55 mpd ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ¬ 𝑈 <s 𝑤 )
57 41 56 jca ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑤 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑤 ) )
58 51 52 53 57 17 syl112anc ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑤 ↾ dom 𝑇 ) = 𝑇 )
59 simpl3r ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → ( 𝑈 ↾ dom 𝑇 ) = 𝑇 )
60 59 adantr ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑈 ↾ dom 𝑇 ) = 𝑇 )
61 58 60 eqtr4d ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑤 ↾ dom 𝑇 ) = ( 𝑈 ↾ dom 𝑇 ) )
62 simplr ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑈 ‘ dom 𝑇 ) = ∅ )
63 nogt01o ⊢ ( ( ( 𝑤 ∈ No ∧ 𝑈 ∈ No ∧ dom 𝑇 ∈ On ) ∧ ( ( 𝑤 ↾ dom 𝑇 ) = ( 𝑈 ↾ dom 𝑇 ) ∧ 𝑤 <s 𝑈 ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → ( 𝑤 ‘ dom 𝑇 ) = 1o )
64 42 44 50 61 54 62 63 syl321anc ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑤 ‘ dom 𝑇 ) = 1o )
65 64 expr ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ 𝑤 ∈ 𝐵 ) → ( 𝑤 <s 𝑈 → ( 𝑤 ‘ dom 𝑇 ) = 1o ) )
66 65 ancld ⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ 𝑤 ∈ 𝐵 ) → ( 𝑤 <s 𝑈 → ( 𝑤 <s 𝑈 ∧ ( 𝑤 ‘ dom 𝑇 ) = 1o ) ) )
67 66 reximdva ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → ( ∃ 𝑤 ∈ 𝐵 𝑤 <s 𝑈 → ∃ 𝑤 ∈ 𝐵 ( 𝑤 <s 𝑈 ∧ ( 𝑤 ‘ dom 𝑇 ) = 1o ) ) )
68 38 67 mpd ⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → ∃ 𝑤 ∈ 𝐵 ( 𝑤 <s 𝑈 ∧ ( 𝑤 ‘ dom 𝑇 ) = 1o ) )
69 25 68 mtand ⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) → ¬ ( 𝑈 ‘ dom 𝑇 ) = ∅ )
70 69 neqned ⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) → ( 𝑈 ‘ dom 𝑇 ) ≠ ∅ )