Metamath Proof Explorer

Theorem nogesgn1o

Description: Given A greater than or equal to B , equal to B up to X , and A ( X ) = 1o , then B ( X ) = 1o . (Contributed by Scott Fenton, 9-Aug-2024)

Ref Expression
Assertion nogesgn1o ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ¬ 𝐴 <s 𝐵 ) → ( 𝐵 ‘ 𝑋 ) = 1o )

Proof

Step Hyp Ref Expression
1 simpl2 ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ) → 𝐵 ∈ No )
2 nofv ⊢ ( 𝐵 ∈ No → ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) )
3 1 2 syl ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ) → ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) )
4 3orel2 ⊢ ( ¬ ( 𝐵 ‘ 𝑋 ) = 1o → ( ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 1o ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) → ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) )
5 3 4 syl5com ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ) → ( ¬ ( 𝐵 ‘ 𝑋 ) = 1o → ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) )
6 simp13 ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → 𝑋 ∈ On )
7 fveq1 ⊢ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) → ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) = ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) )
8 7 adantr ⊢ ( ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ 𝑦 ∈ 𝑋 ) → ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) = ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) )
9 simpr ⊢ ( ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ 𝑦 ∈ 𝑋 ) → 𝑦 ∈ 𝑋 )
10 9 fvresd ⊢ ( ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ 𝑦 ∈ 𝑋 ) → ( ( 𝐴 ↾ 𝑋 ) ‘ 𝑦 ) = ( 𝐴 ‘ 𝑦 ) )
11 9 fvresd ⊢ ( ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ 𝑦 ∈ 𝑋 ) → ( ( 𝐵 ↾ 𝑋 ) ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) )
12 8 10 11 3eqtr3d ⊢ ( ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ 𝑦 ∈ 𝑋 ) → ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) )
13 12 ralrimiva ⊢ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) → ∀ 𝑦 ∈ 𝑋 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) )
14 13 adantr ⊢ ( ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) → ∀ 𝑦 ∈ 𝑋 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) )
15 14 3ad2ant2 ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → ∀ 𝑦 ∈ 𝑋 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) )
16 simp2r ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → ( 𝐴 ‘ 𝑋 ) = 1o )
17 simp3 ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) )
18 16 17 jca ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) )
19 andi ⊢ ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) ↔ ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ) )
20 18 19 sylib ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ) )
21 3mix1 ⊢ ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = ∅ ) → ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ∨ ( ( 𝐴 ‘ 𝑋 ) = ∅ ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ) )
22 3mix2 ⊢ ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) → ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ∨ ( ( 𝐴 ‘ 𝑋 ) = ∅ ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ) )
23 21 22 jaoi ⊢ ( ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ∨ ( ( 𝐴 ‘ 𝑋 ) = ∅ ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ) )
24 20 23 syl ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ∨ ( ( 𝐴 ‘ 𝑋 ) = ∅ ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ) )
25 fvex ⊢ ( 𝐴 ‘ 𝑋 ) ∈ V
26 fvex ⊢ ( 𝐵 ‘ 𝑋 ) ∈ V
27 25 26 brtp ⊢ ( ( 𝐴 ‘ 𝑋 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑋 ) ↔ ( ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = ∅ ) ∨ ( ( 𝐴 ‘ 𝑋 ) = 1o ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ∨ ( ( 𝐴 ‘ 𝑋 ) = ∅ ∧ ( 𝐵 ‘ 𝑋 ) = 2o ) ) )
28 24 27 sylibr ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → ( 𝐴 ‘ 𝑋 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑋 ) )
29 raleq ⊢ ( 𝑥 = 𝑋 → ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ↔ ∀ 𝑦 ∈ 𝑋 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ) )
30 fveq2 ⊢ ( 𝑥 = 𝑋 → ( 𝐴 ‘ 𝑥 ) = ( 𝐴 ‘ 𝑋 ) )
31 fveq2 ⊢ ( 𝑥 = 𝑋 → ( 𝐵 ‘ 𝑥 ) = ( 𝐵 ‘ 𝑋 ) )
32 30 31 breq12d ⊢ ( 𝑥 = 𝑋 → ( ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ↔ ( 𝐴 ‘ 𝑋 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑋 ) ) )
33 29 32 anbi12d ⊢ ( 𝑥 = 𝑋 → ( ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ) ↔ ( ∀ 𝑦 ∈ 𝑋 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑋 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑋 ) ) ) )
34 33 rspcev ⊢ ( ( 𝑋 ∈ On ∧ ( ∀ 𝑦 ∈ 𝑋 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑋 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑋 ) ) ) → ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ) )
35 6 15 28 34 syl12anc ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ) )
36 simp11 ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → 𝐴 ∈ No )
37 simp12 ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → 𝐵 ∈ No )
38 ltsval ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 <s 𝐵 ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ) ) )
39 36 37 38 syl2anc ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → ( 𝐴 <s 𝐵 ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦 ∈ 𝑥 ( 𝐴 ‘ 𝑦 ) = ( 𝐵 ‘ 𝑦 ) ∧ ( 𝐴 ‘ 𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 ‘ 𝑥 ) ) ) )
40 35 39 mpbird ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) ) → 𝐴 <s 𝐵 )
41 40 3expia ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ) → ( ( ( 𝐵 ‘ 𝑋 ) = ∅ ∨ ( 𝐵 ‘ 𝑋 ) = 2o ) → 𝐴 <s 𝐵 ) )
42 5 41 syld ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ) → ( ¬ ( 𝐵 ‘ 𝑋 ) = 1o → 𝐴 <s 𝐵 ) )
43 42 con1d ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ) → ( ¬ 𝐴 <s 𝐵 → ( 𝐵 ‘ 𝑋 ) = 1o ) )
44 43 3impia ⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On ) ∧ ( ( 𝐴 ↾ 𝑋 ) = ( 𝐵 ↾ 𝑋 ) ∧ ( 𝐴 ‘ 𝑋 ) = 1o ) ∧ ¬ 𝐴 <s 𝐵 ) → ( 𝐵 ‘ 𝑋 ) = 1o )