Metamath Proof Explorer


Theorem nosupbnd1lem1

Description: Lemma for nosupbnd1 . Establish a soft upper bound. (Contributed by Scott Fenton, 5-Dec-2021)

Ref Expression
Hypothesis nosupbnd1.1 𝑆 = if ( ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 , ( ( 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { ⟨ dom ( 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } ) , ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥𝑢𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) ) )
Assertion nosupbnd1lem1 ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) → ¬ 𝑆 <s ( 𝑈 ↾ dom 𝑆 ) )

Proof

Step Hyp Ref Expression
1 nosupbnd1.1 𝑆 = if ( ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 , ( ( 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { ⟨ dom ( 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } ) , ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥𝑢𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) ) )
2 simp2l ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) → 𝐴 No )
3 simp3 ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) → 𝑈𝐴 )
4 2 3 sseldd ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) → 𝑈 No )
5 1 nosupno ( ( 𝐴 No 𝐴 ∈ V ) → 𝑆 No )
6 5 3ad2ant2 ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) → 𝑆 No )
7 nodmon ( 𝑆 No → dom 𝑆 ∈ On )
8 6 7 syl ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) → dom 𝑆 ∈ On )
9 noreson ( ( 𝑈 No ∧ dom 𝑆 ∈ On ) → ( 𝑈 ↾ dom 𝑆 ) ∈ No )
10 4 8 9 syl2anc ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) → ( 𝑈 ↾ dom 𝑆 ) ∈ No )
11 dmres dom ( 𝑈 ↾ dom 𝑆 ) = ( dom 𝑆 ∩ dom 𝑈 )
12 inss1 ( dom 𝑆 ∩ dom 𝑈 ) ⊆ dom 𝑆
13 11 12 eqsstri dom ( 𝑈 ↾ dom 𝑆 ) ⊆ dom 𝑆
14 13 a1i ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) → dom ( 𝑈 ↾ dom 𝑆 ) ⊆ dom 𝑆 )
15 ssidd ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) → dom 𝑆 ⊆ dom 𝑆 )
16 iffalse ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → if ( ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 , ( ( 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { ⟨ dom ( 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } ) , ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥𝑢𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) ) ) = ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥𝑢𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) ) )
17 1 16 syl5eq ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦𝑆 = ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥𝑢𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) ) )
18 17 dmeqd ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = dom ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥𝑢𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) ) )
19 iotaex ( ℩ 𝑥𝑢𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) ∈ V
20 eqid ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥𝑢𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) ) = ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥𝑢𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) )
21 19 20 dmmpti dom ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥𝑢𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢𝑔 ) = 𝑥 ) ) ) = { 𝑦 ∣ ∃ 𝑢𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) }
22 18 21 eqtrdi ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = { 𝑦 ∣ ∃ 𝑢𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } )
23 22 eleq2d ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → ( ∈ dom 𝑆 ∈ { 𝑦 ∣ ∃ 𝑢𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ) )
24 vex ∈ V
25 eleq1w ( 𝑦 = → ( 𝑦 ∈ dom 𝑢 ∈ dom 𝑢 ) )
26 suceq ( 𝑦 = → suc 𝑦 = suc )
27 26 reseq2d ( 𝑦 = → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑢 ↾ suc ) )
28 26 reseq2d ( 𝑦 = → ( 𝑣 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc ) )
29 27 28 eqeq12d ( 𝑦 = → ( ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ↔ ( 𝑢 ↾ suc ) = ( 𝑣 ↾ suc ) ) )
30 29 imbi2d ( 𝑦 = → ( ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ↔ ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) )
31 30 ralbidv ( 𝑦 = → ( ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ↔ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) )
32 25 31 anbi12d ( 𝑦 = → ( ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) ↔ ( ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) )
33 32 rexbidv ( 𝑦 = → ( ∃ 𝑢𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) ↔ ∃ 𝑢𝐴 ( ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) )
34 dmeq ( 𝑢 = 𝑝 → dom 𝑢 = dom 𝑝 )
35 34 eleq2d ( 𝑢 = 𝑝 → ( ∈ dom 𝑢 ∈ dom 𝑝 ) )
36 breq2 ( 𝑢 = 𝑝 → ( 𝑣 <s 𝑢𝑣 <s 𝑝 ) )
37 36 notbid ( 𝑢 = 𝑝 → ( ¬ 𝑣 <s 𝑢 ↔ ¬ 𝑣 <s 𝑝 ) )
38 reseq1 ( 𝑢 = 𝑝 → ( 𝑢 ↾ suc ) = ( 𝑝 ↾ suc ) )
39 38 eqeq1d ( 𝑢 = 𝑝 → ( ( 𝑢 ↾ suc ) = ( 𝑣 ↾ suc ) ↔ ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) )
40 37 39 imbi12d ( 𝑢 = 𝑝 → ( ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc ) = ( 𝑣 ↾ suc ) ) ↔ ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) )
41 40 ralbidv ( 𝑢 = 𝑝 → ( ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc ) = ( 𝑣 ↾ suc ) ) ↔ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) )
42 35 41 anbi12d ( 𝑢 = 𝑝 → ( ( ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ↔ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) )
43 42 cbvrexvw ( ∃ 𝑢𝐴 ( ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ↔ ∃ 𝑝𝐴 ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) )
44 33 43 bitrdi ( 𝑦 = → ( ∃ 𝑢𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) ↔ ∃ 𝑝𝐴 ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) )
45 24 44 elab ( ∈ { 𝑦 ∣ ∃ 𝑢𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↔ ∃ 𝑝𝐴 ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) )
46 23 45 bitrdi ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → ( ∈ dom 𝑆 ↔ ∃ 𝑝𝐴 ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) )
47 46 3ad2ant1 ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) → ( ∈ dom 𝑆 ↔ ∃ 𝑝𝐴 ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) )
48 simpl1 ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 )
49 simpl2 ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → ( 𝐴 No 𝐴 ∈ V ) )
50 simprl ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → 𝑝𝐴 )
51 simprrl ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → ∈ dom 𝑝 )
52 simprrr ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) )
53 1 nosupres ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ ( 𝑝𝐴 ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) → ( 𝑆 ↾ suc ) = ( 𝑝 ↾ suc ) )
54 48 49 50 51 52 53 syl113anc ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → ( 𝑆 ↾ suc ) = ( 𝑝 ↾ suc ) )
55 simpl2l ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → 𝐴 No )
56 55 50 sseldd ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → 𝑝 No )
57 4 adantr ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → 𝑈 No )
58 sltso <s Or No
59 soasym ( ( <s Or No ∧ ( 𝑝 No 𝑈 No ) ) → ( 𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝 ) )
60 58 59 mpan ( ( 𝑝 No 𝑈 No ) → ( 𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝 ) )
61 56 57 60 syl2anc ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → ( 𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝 ) )
62 simpl3 ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → 𝑈𝐴 )
63 breq1 ( 𝑣 = 𝑈 → ( 𝑣 <s 𝑝𝑈 <s 𝑝 ) )
64 63 notbid ( 𝑣 = 𝑈 → ( ¬ 𝑣 <s 𝑝 ↔ ¬ 𝑈 <s 𝑝 ) )
65 reseq1 ( 𝑣 = 𝑈 → ( 𝑣 ↾ suc ) = ( 𝑈 ↾ suc ) )
66 65 eqeq2d ( 𝑣 = 𝑈 → ( ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ↔ ( 𝑝 ↾ suc ) = ( 𝑈 ↾ suc ) ) )
67 64 66 imbi12d ( 𝑣 = 𝑈 → ( ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ↔ ( ¬ 𝑈 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑈 ↾ suc ) ) ) )
68 67 rspcv ( 𝑈𝐴 → ( ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) → ( ¬ 𝑈 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑈 ↾ suc ) ) ) )
69 62 52 68 sylc ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → ( ¬ 𝑈 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑈 ↾ suc ) ) )
70 61 69 syld ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → ( 𝑝 <s 𝑈 → ( 𝑝 ↾ suc ) = ( 𝑈 ↾ suc ) ) )
71 70 imp ( ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) ∧ 𝑝 <s 𝑈 ) → ( 𝑝 ↾ suc ) = ( 𝑈 ↾ suc ) )
72 nodmon ( 𝑝 No → dom 𝑝 ∈ On )
73 56 72 syl ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → dom 𝑝 ∈ On )
74 onelon ( ( dom 𝑝 ∈ On ∧ ∈ dom 𝑝 ) → ∈ On )
75 73 51 74 syl2anc ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → ∈ On )
76 sucelon ( ∈ On ↔ suc ∈ On )
77 75 76 sylib ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → suc ∈ On )
78 noreson ( ( 𝑈 No ∧ suc ∈ On ) → ( 𝑈 ↾ suc ) ∈ No )
79 57 77 78 syl2anc ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → ( 𝑈 ↾ suc ) ∈ No )
80 sonr ( ( <s Or No ∧ ( 𝑈 ↾ suc ) ∈ No ) → ¬ ( 𝑈 ↾ suc ) <s ( 𝑈 ↾ suc ) )
81 58 80 mpan ( ( 𝑈 ↾ suc ) ∈ No → ¬ ( 𝑈 ↾ suc ) <s ( 𝑈 ↾ suc ) )
82 79 81 syl ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → ¬ ( 𝑈 ↾ suc ) <s ( 𝑈 ↾ suc ) )
83 82 adantr ( ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) ∧ 𝑝 <s 𝑈 ) → ¬ ( 𝑈 ↾ suc ) <s ( 𝑈 ↾ suc ) )
84 71 83 eqnbrtrd ( ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) ∧ 𝑝 <s 𝑈 ) → ¬ ( 𝑝 ↾ suc ) <s ( 𝑈 ↾ suc ) )
85 84 ex ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → ( 𝑝 <s 𝑈 → ¬ ( 𝑝 ↾ suc ) <s ( 𝑈 ↾ suc ) ) )
86 sltres ( ( 𝑝 No 𝑈 No ∧ suc ∈ On ) → ( ( 𝑝 ↾ suc ) <s ( 𝑈 ↾ suc ) → 𝑝 <s 𝑈 ) )
87 56 57 77 86 syl3anc ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → ( ( 𝑝 ↾ suc ) <s ( 𝑈 ↾ suc ) → 𝑝 <s 𝑈 ) )
88 87 con3d ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → ( ¬ 𝑝 <s 𝑈 → ¬ ( 𝑝 ↾ suc ) <s ( 𝑈 ↾ suc ) ) )
89 85 88 pm2.61d ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → ¬ ( 𝑝 ↾ suc ) <s ( 𝑈 ↾ suc ) )
90 54 89 eqnbrtrd ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ( 𝑝𝐴 ∧ ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) ) ) → ¬ ( 𝑆 ↾ suc ) <s ( 𝑈 ↾ suc ) )
91 90 rexlimdvaa ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) → ( ∃ 𝑝𝐴 ( ∈ dom 𝑝 ∧ ∀ 𝑣𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ) = ( 𝑣 ↾ suc ) ) ) → ¬ ( 𝑆 ↾ suc ) <s ( 𝑈 ↾ suc ) ) )
92 47 91 sylbid ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) → ( ∈ dom 𝑆 → ¬ ( 𝑆 ↾ suc ) <s ( 𝑈 ↾ suc ) ) )
93 92 imp ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ∈ dom 𝑆 ) → ¬ ( 𝑆 ↾ suc ) <s ( 𝑈 ↾ suc ) )
94 nodmord ( 𝑆 No → Ord dom 𝑆 )
95 ordsucss ( Ord dom 𝑆 → ( ∈ dom 𝑆 → suc ⊆ dom 𝑆 ) )
96 6 94 95 3syl ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) → ( ∈ dom 𝑆 → suc ⊆ dom 𝑆 ) )
97 96 imp ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ∈ dom 𝑆 ) → suc ⊆ dom 𝑆 )
98 97 resabs1d ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ∈ dom 𝑆 ) → ( ( 𝑈 ↾ dom 𝑆 ) ↾ suc ) = ( 𝑈 ↾ suc ) )
99 98 breq2d ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ∈ dom 𝑆 ) → ( ( 𝑆 ↾ suc ) <s ( ( 𝑈 ↾ dom 𝑆 ) ↾ suc ) ↔ ( 𝑆 ↾ suc ) <s ( 𝑈 ↾ suc ) ) )
100 93 99 mtbird ( ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) ∧ ∈ dom 𝑆 ) → ¬ ( 𝑆 ↾ suc ) <s ( ( 𝑈 ↾ dom 𝑆 ) ↾ suc ) )
101 100 ralrimiva ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) → ∀ ∈ dom 𝑆 ¬ ( 𝑆 ↾ suc ) <s ( ( 𝑈 ↾ dom 𝑆 ) ↾ suc ) )
102 noresle ( ( ( ( 𝑈 ↾ dom 𝑆 ) ∈ No 𝑆 No ) ∧ ( dom ( 𝑈 ↾ dom 𝑆 ) ⊆ dom 𝑆 ∧ dom 𝑆 ⊆ dom 𝑆 ∧ ∀ ∈ dom 𝑆 ¬ ( 𝑆 ↾ suc ) <s ( ( 𝑈 ↾ dom 𝑆 ) ↾ suc ) ) ) → ¬ 𝑆 <s ( 𝑈 ↾ dom 𝑆 ) )
103 10 6 14 15 101 102 syl23anc ( ( ¬ ∃ 𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 No 𝐴 ∈ V ) ∧ 𝑈𝐴 ) → ¬ 𝑆 <s ( 𝑈 ↾ dom 𝑆 ) )