Metamath Proof Explorer


Theorem nosepnelem

Description: Lemma for nosepne . (Contributed by Scott Fenton, 24-Nov-2021)

Ref Expression
Assertion nosepnelem ( ( 𝐴 No 𝐵 No 𝐴 <s 𝐵 ) → ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) ≠ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) )

Proof

Step Hyp Ref Expression
1 sltval2 ( ( 𝐴 No 𝐵 No ) → ( 𝐴 <s 𝐵 ↔ ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) ) )
2 fvex ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) ∈ V
3 fvex ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) ∈ V
4 2 3 brtp ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) ↔ ( ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 1o ∧ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = ∅ ) ∨ ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 1o ∧ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 2o ) ∨ ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = ∅ ∧ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 2o ) ) )
5 1n0 1o ≠ ∅
6 simpl ( ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 1o ∧ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = ∅ ) → ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 1o )
7 simpr ( ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 1o ∧ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = ∅ ) → ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = ∅ )
8 6 7 neeq12d ( ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 1o ∧ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = ∅ ) → ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) ≠ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) ↔ 1o ≠ ∅ ) )
9 5 8 mpbiri ( ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 1o ∧ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = ∅ ) → ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) ≠ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) )
10 df-2o 2o = suc 1o
11 df-1o 1o = suc ∅
12 10 11 eqeq12i ( 2o = 1o ↔ suc 1o = suc ∅ )
13 1on 1o ∈ On
14 0elon ∅ ∈ On
15 suc11 ( ( 1o ∈ On ∧ ∅ ∈ On ) → ( suc 1o = suc ∅ ↔ 1o = ∅ ) )
16 13 14 15 mp2an ( suc 1o = suc ∅ ↔ 1o = ∅ )
17 12 16 bitri ( 2o = 1o ↔ 1o = ∅ )
18 17 necon3bii ( 2o ≠ 1o ↔ 1o ≠ ∅ )
19 5 18 mpbir 2o ≠ 1o
20 19 necomi 1o ≠ 2o
21 simpl ( ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 1o ∧ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 2o ) → ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 1o )
22 simpr ( ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 1o ∧ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 2o ) → ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 2o )
23 21 22 neeq12d ( ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 1o ∧ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 2o ) → ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) ≠ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) ↔ 1o ≠ 2o ) )
24 20 23 mpbiri ( ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 1o ∧ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 2o ) → ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) ≠ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) )
25 2on 2o ∈ On
26 25 elexi 2o ∈ V
27 26 prid2 2o ∈ { 1o , 2o }
28 27 nosgnn0i ∅ ≠ 2o
29 simpl ( ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = ∅ ∧ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 2o ) → ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = ∅ )
30 simpr ( ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = ∅ ∧ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 2o ) → ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 2o )
31 29 30 neeq12d ( ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = ∅ ∧ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 2o ) → ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) ≠ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) ↔ ∅ ≠ 2o ) )
32 28 31 mpbiri ( ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = ∅ ∧ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 2o ) → ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) ≠ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) )
33 9 24 32 3jaoi ( ( ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 1o ∧ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = ∅ ) ∨ ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 1o ∧ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 2o ) ∨ ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = ∅ ∧ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) = 2o ) ) → ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) ≠ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) )
34 4 33 sylbi ( ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) { ⟨ 1o , ∅ ⟩ , ⟨ 1o , 2o ⟩ , ⟨ ∅ , 2o ⟩ } ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) → ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) ≠ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) )
35 1 34 syl6bi ( ( 𝐴 No 𝐵 No ) → ( 𝐴 <s 𝐵 → ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) ≠ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) ) )
36 35 3impia ( ( 𝐴 No 𝐵 No 𝐴 <s 𝐵 ) → ( 𝐴 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) ≠ ( 𝐵 { 𝑥 ∈ On ∣ ( 𝐴𝑥 ) ≠ ( 𝐵𝑥 ) } ) )