Metamath Proof Explorer


Theorem nolesgn2o

Description: Given A less than or equal to B , equal to B up to X , and A ( X ) = 2o , then B ( X ) = 2o . (Contributed by Scott Fenton, 6-Dec-2021)

Ref Expression
Assertion nolesgn2o ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ∧ ¬ 𝐵 <s 𝐴 ) → ( 𝐵𝑋 ) = 2o )

Proof

Step Hyp Ref Expression
1 simpl2 ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ) → 𝐵 No )
2 nofv ( 𝐵 No → ( ( 𝐵𝑋 ) = ∅ ∨ ( 𝐵𝑋 ) = 1o ∨ ( 𝐵𝑋 ) = 2o ) )
3 1 2 syl ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ) → ( ( 𝐵𝑋 ) = ∅ ∨ ( 𝐵𝑋 ) = 1o ∨ ( 𝐵𝑋 ) = 2o ) )
4 3orel3 ( ¬ ( 𝐵𝑋 ) = 2o → ( ( ( 𝐵𝑋 ) = ∅ ∨ ( 𝐵𝑋 ) = 1o ∨ ( 𝐵𝑋 ) = 2o ) → ( ( 𝐵𝑋 ) = ∅ ∨ ( 𝐵𝑋 ) = 1o ) ) )
5 3 4 syl5com ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ) → ( ¬ ( 𝐵𝑋 ) = 2o → ( ( 𝐵𝑋 ) = ∅ ∨ ( 𝐵𝑋 ) = 1o ) ) )
6 simp13 ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ∧ ( ( 𝐵𝑋 ) = ∅ ∨ ( 𝐵𝑋 ) = 1o ) ) → 𝑋 ∈ On )
7 fveq1 ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) → ( ( 𝐴𝑋 ) ‘ 𝑦 ) = ( ( 𝐵𝑋 ) ‘ 𝑦 ) )
8 7 eqcomd ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) → ( ( 𝐵𝑋 ) ‘ 𝑦 ) = ( ( 𝐴𝑋 ) ‘ 𝑦 ) )
9 8 adantr ( ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ 𝑦𝑋 ) → ( ( 𝐵𝑋 ) ‘ 𝑦 ) = ( ( 𝐴𝑋 ) ‘ 𝑦 ) )
10 simpr ( ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ 𝑦𝑋 ) → 𝑦𝑋 )
11 10 fvresd ( ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ 𝑦𝑋 ) → ( ( 𝐵𝑋 ) ‘ 𝑦 ) = ( 𝐵𝑦 ) )
12 10 fvresd ( ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ 𝑦𝑋 ) → ( ( 𝐴𝑋 ) ‘ 𝑦 ) = ( 𝐴𝑦 ) )
13 9 11 12 3eqtr3d ( ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ 𝑦𝑋 ) → ( 𝐵𝑦 ) = ( 𝐴𝑦 ) )
14 13 ralrimiva ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) → ∀ 𝑦𝑋 ( 𝐵𝑦 ) = ( 𝐴𝑦 ) )
15 14 adantr ( ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) → ∀ 𝑦𝑋 ( 𝐵𝑦 ) = ( 𝐴𝑦 ) )
16 15 3ad2ant2 ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ∧ ( ( 𝐵𝑋 ) = ∅ ∨ ( 𝐵𝑋 ) = 1o ) ) → ∀ 𝑦𝑋 ( 𝐵𝑦 ) = ( 𝐴𝑦 ) )
17 simprr ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ) → ( 𝐴𝑋 ) = 2o )
18 17 a1d ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ) → ( ( 𝐵𝑋 ) = ∅ → ( 𝐴𝑋 ) = 2o ) )
19 18 ancld ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ) → ( ( 𝐵𝑋 ) = ∅ → ( ( 𝐵𝑋 ) = ∅ ∧ ( 𝐴𝑋 ) = 2o ) ) )
20 17 a1d ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ) → ( ( 𝐵𝑋 ) = 1o → ( 𝐴𝑋 ) = 2o ) )
21 20 ancld ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ) → ( ( 𝐵𝑋 ) = 1o → ( ( 𝐵𝑋 ) = 1o ∧ ( 𝐴𝑋 ) = 2o ) ) )
22 19 21 orim12d ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ) → ( ( ( 𝐵𝑋 ) = ∅ ∨ ( 𝐵𝑋 ) = 1o ) → ( ( ( 𝐵𝑋 ) = ∅ ∧ ( 𝐴𝑋 ) = 2o ) ∨ ( ( 𝐵𝑋 ) = 1o ∧ ( 𝐴𝑋 ) = 2o ) ) ) )
23 22 3impia ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ∧ ( ( 𝐵𝑋 ) = ∅ ∨ ( 𝐵𝑋 ) = 1o ) ) → ( ( ( 𝐵𝑋 ) = ∅ ∧ ( 𝐴𝑋 ) = 2o ) ∨ ( ( 𝐵𝑋 ) = 1o ∧ ( 𝐴𝑋 ) = 2o ) ) )
24 3mix3 ( ( ( 𝐵𝑋 ) = ∅ ∧ ( 𝐴𝑋 ) = 2o ) → ( ( ( 𝐵𝑋 ) = 1o ∧ ( 𝐴𝑋 ) = ∅ ) ∨ ( ( 𝐵𝑋 ) = 1o ∧ ( 𝐴𝑋 ) = 2o ) ∨ ( ( 𝐵𝑋 ) = ∅ ∧ ( 𝐴𝑋 ) = 2o ) ) )
25 3mix2 ( ( ( 𝐵𝑋 ) = 1o ∧ ( 𝐴𝑋 ) = 2o ) → ( ( ( 𝐵𝑋 ) = 1o ∧ ( 𝐴𝑋 ) = ∅ ) ∨ ( ( 𝐵𝑋 ) = 1o ∧ ( 𝐴𝑋 ) = 2o ) ∨ ( ( 𝐵𝑋 ) = ∅ ∧ ( 𝐴𝑋 ) = 2o ) ) )
26 24 25 jaoi ( ( ( ( 𝐵𝑋 ) = ∅ ∧ ( 𝐴𝑋 ) = 2o ) ∨ ( ( 𝐵𝑋 ) = 1o ∧ ( 𝐴𝑋 ) = 2o ) ) → ( ( ( 𝐵𝑋 ) = 1o ∧ ( 𝐴𝑋 ) = ∅ ) ∨ ( ( 𝐵𝑋 ) = 1o ∧ ( 𝐴𝑋 ) = 2o ) ∨ ( ( 𝐵𝑋 ) = ∅ ∧ ( 𝐴𝑋 ) = 2o ) ) )
27 23 26 syl ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ∧ ( ( 𝐵𝑋 ) = ∅ ∨ ( 𝐵𝑋 ) = 1o ) ) → ( ( ( 𝐵𝑋 ) = 1o ∧ ( 𝐴𝑋 ) = ∅ ) ∨ ( ( 𝐵𝑋 ) = 1o ∧ ( 𝐴𝑋 ) = 2o ) ∨ ( ( 𝐵𝑋 ) = ∅ ∧ ( 𝐴𝑋 ) = 2o ) ) )
28 fvex ( 𝐵𝑋 ) ∈ V
29 fvex ( 𝐴𝑋 ) ∈ V
30 28 29 brtp ( ( 𝐵𝑋 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐴𝑋 ) ↔ ( ( ( 𝐵𝑋 ) = 1o ∧ ( 𝐴𝑋 ) = ∅ ) ∨ ( ( 𝐵𝑋 ) = 1o ∧ ( 𝐴𝑋 ) = 2o ) ∨ ( ( 𝐵𝑋 ) = ∅ ∧ ( 𝐴𝑋 ) = 2o ) ) )
31 27 30 sylibr ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ∧ ( ( 𝐵𝑋 ) = ∅ ∨ ( 𝐵𝑋 ) = 1o ) ) → ( 𝐵𝑋 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐴𝑋 ) )
32 raleq ( 𝑥 = 𝑋 → ( ∀ 𝑦𝑥 ( 𝐵𝑦 ) = ( 𝐴𝑦 ) ↔ ∀ 𝑦𝑋 ( 𝐵𝑦 ) = ( 𝐴𝑦 ) ) )
33 fveq2 ( 𝑥 = 𝑋 → ( 𝐵𝑥 ) = ( 𝐵𝑋 ) )
34 fveq2 ( 𝑥 = 𝑋 → ( 𝐴𝑥 ) = ( 𝐴𝑋 ) )
35 33 34 breq12d ( 𝑥 = 𝑋 → ( ( 𝐵𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐴𝑥 ) ↔ ( 𝐵𝑋 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐴𝑋 ) ) )
36 32 35 anbi12d ( 𝑥 = 𝑋 → ( ( ∀ 𝑦𝑥 ( 𝐵𝑦 ) = ( 𝐴𝑦 ) ∧ ( 𝐵𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐴𝑥 ) ) ↔ ( ∀ 𝑦𝑋 ( 𝐵𝑦 ) = ( 𝐴𝑦 ) ∧ ( 𝐵𝑋 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐴𝑋 ) ) ) )
37 36 rspcev ( ( 𝑋 ∈ On ∧ ( ∀ 𝑦𝑋 ( 𝐵𝑦 ) = ( 𝐴𝑦 ) ∧ ( 𝐵𝑋 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐴𝑋 ) ) ) → ∃ 𝑥 ∈ On ( ∀ 𝑦𝑥 ( 𝐵𝑦 ) = ( 𝐴𝑦 ) ∧ ( 𝐵𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐴𝑥 ) ) )
38 6 16 31 37 syl12anc ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ∧ ( ( 𝐵𝑋 ) = ∅ ∨ ( 𝐵𝑋 ) = 1o ) ) → ∃ 𝑥 ∈ On ( ∀ 𝑦𝑥 ( 𝐵𝑦 ) = ( 𝐴𝑦 ) ∧ ( 𝐵𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐴𝑥 ) ) )
39 simp12 ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ∧ ( ( 𝐵𝑋 ) = ∅ ∨ ( 𝐵𝑋 ) = 1o ) ) → 𝐵 No )
40 simp11 ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ∧ ( ( 𝐵𝑋 ) = ∅ ∨ ( 𝐵𝑋 ) = 1o ) ) → 𝐴 No )
41 sltval ( ( 𝐵 No 𝐴 No ) → ( 𝐵 <s 𝐴 ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦𝑥 ( 𝐵𝑦 ) = ( 𝐴𝑦 ) ∧ ( 𝐵𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐴𝑥 ) ) ) )
42 39 40 41 syl2anc ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ∧ ( ( 𝐵𝑋 ) = ∅ ∨ ( 𝐵𝑋 ) = 1o ) ) → ( 𝐵 <s 𝐴 ↔ ∃ 𝑥 ∈ On ( ∀ 𝑦𝑥 ( 𝐵𝑦 ) = ( 𝐴𝑦 ) ∧ ( 𝐵𝑥 ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐴𝑥 ) ) ) )
43 38 42 mpbird ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ∧ ( ( 𝐵𝑋 ) = ∅ ∨ ( 𝐵𝑋 ) = 1o ) ) → 𝐵 <s 𝐴 )
44 43 3expia ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ) → ( ( ( 𝐵𝑋 ) = ∅ ∨ ( 𝐵𝑋 ) = 1o ) → 𝐵 <s 𝐴 ) )
45 5 44 syld ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ) → ( ¬ ( 𝐵𝑋 ) = 2o𝐵 <s 𝐴 ) )
46 45 con1d ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ) → ( ¬ 𝐵 <s 𝐴 → ( 𝐵𝑋 ) = 2o ) )
47 46 3impia ( ( ( 𝐴 No 𝐵 No 𝑋 ∈ On ) ∧ ( ( 𝐴𝑋 ) = ( 𝐵𝑋 ) ∧ ( 𝐴𝑋 ) = 2o ) ∧ ¬ 𝐵 <s 𝐴 ) → ( 𝐵𝑋 ) = 2o )