Metamath Proof Explorer


Theorem frege133

Description: If the procedure R is single-valued and if M and Y follow X in the R -sequence, then Y belongs to the R -sequence beginning with M or precedes M in the R -sequence. Proposition 133 of Frege1879 p. 86. (Contributed by RP, 9-Jul-2020) (Proof modification is discouraged.)

Ref Expression
Hypotheses frege133.x 𝑋𝑈
frege133.y 𝑌𝑉
frege133.m 𝑀𝑊
frege133.r 𝑅𝑆
Assertion frege133 ( Fun 𝑅 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) )

Proof

Step Hyp Ref Expression
1 frege133.x 𝑋𝑈
2 frege133.y 𝑌𝑉
3 frege133.m 𝑀𝑊
4 frege133.r 𝑅𝑆
5 fvex ( t+ ‘ 𝑅 ) ∈ V
6 5 cnvex ( t+ ‘ 𝑅 ) ∈ V
7 imaexg ( ( t+ ‘ 𝑅 ) ∈ V → ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∈ V )
8 6 7 ax-mp ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∈ V
9 imaundir ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) = ( ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( I “ { 𝑀 } ) )
10 imaexg ( ( t+ ‘ 𝑅 ) ∈ V → ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∈ V )
11 5 10 ax-mp ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∈ V
12 imai ( I “ { 𝑀 } ) = { 𝑀 }
13 snex { 𝑀 } ∈ V
14 12 13 eqeltri ( I “ { 𝑀 } ) ∈ V
15 11 14 unex ( ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( I “ { 𝑀 } ) ) ∈ V
16 9 15 eqeltri ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ∈ V
17 1 2 4 8 16 frege83 ( 𝑅 hereditary ( ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) → ( 𝑋 ∈ ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌𝑌 ∈ ( ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) ) )
18 3 elexi 𝑀 ∈ V
19 1 elexi 𝑋 ∈ V
20 18 19 elimasn ( 𝑋 ∈ ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ↔ ⟨ 𝑀 , 𝑋 ⟩ ∈ ( t+ ‘ 𝑅 ) )
21 df-br ( 𝑀 ( t+ ‘ 𝑅 ) 𝑋 ↔ ⟨ 𝑀 , 𝑋 ⟩ ∈ ( t+ ‘ 𝑅 ) )
22 18 19 brcnv ( 𝑀 ( t+ ‘ 𝑅 ) 𝑋𝑋 ( t+ ‘ 𝑅 ) 𝑀 )
23 20 21 22 3bitr2i ( 𝑋 ∈ ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ↔ 𝑋 ( t+ ‘ 𝑅 ) 𝑀 )
24 elun ( 𝑌 ∈ ( ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ↔ ( 𝑌 ∈ ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∨ 𝑌 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) )
25 df-or ( ( 𝑌 ∈ ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∨ 𝑌 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ↔ ( ¬ 𝑌 ∈ ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) → 𝑌 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) )
26 2 elexi 𝑌 ∈ V
27 18 26 elimasn ( 𝑌 ∈ ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ↔ ⟨ 𝑀 , 𝑌 ⟩ ∈ ( t+ ‘ 𝑅 ) )
28 df-br ( 𝑀 ( t+ ‘ 𝑅 ) 𝑌 ↔ ⟨ 𝑀 , 𝑌 ⟩ ∈ ( t+ ‘ 𝑅 ) )
29 18 26 brcnv ( 𝑀 ( t+ ‘ 𝑅 ) 𝑌𝑌 ( t+ ‘ 𝑅 ) 𝑀 )
30 27 28 29 3bitr2i ( 𝑌 ∈ ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ↔ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 )
31 30 notbii ( ¬ 𝑌 ∈ ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ↔ ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀 )
32 18 26 elimasn ( 𝑌 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ↔ ⟨ 𝑀 , 𝑌 ⟩ ∈ ( ( t+ ‘ 𝑅 ) ∪ I ) )
33 df-br ( 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ↔ ⟨ 𝑀 , 𝑌 ⟩ ∈ ( ( t+ ‘ 𝑅 ) ∪ I ) )
34 32 33 bitr4i ( 𝑌 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ↔ 𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 )
35 31 34 imbi12i ( ( ¬ 𝑌 ∈ ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) → 𝑌 ∈ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ↔ ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) )
36 24 25 35 3bitri ( 𝑌 ∈ ( ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ↔ ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) )
37 36 imbi2i ( ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌𝑌 ∈ ( ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) ↔ ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) )
38 23 37 imbi12i ( ( 𝑋 ∈ ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌𝑌 ∈ ( ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) ) ↔ ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) )
39 38 imbi2i ( ( 𝑅 hereditary ( ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) → ( 𝑋 ∈ ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌𝑌 ∈ ( ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) ) ) ↔ ( 𝑅 hereditary ( ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) ) )
40 3 4 frege132 ( ( 𝑅 hereditary ( ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) ) → ( Fun 𝑅 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) ) )
41 39 40 sylbi ( ( 𝑅 hereditary ( ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) → ( 𝑋 ∈ ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌𝑌 ∈ ( ( ( t+ ‘ 𝑅 ) “ { 𝑀 } ) ∪ ( ( ( t+ ‘ 𝑅 ) ∪ I ) “ { 𝑀 } ) ) ) ) ) → ( Fun 𝑅 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) ) )
42 17 41 ax-mp ( Fun 𝑅 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑀 → ( 𝑋 ( t+ ‘ 𝑅 ) 𝑌 → ( ¬ 𝑌 ( t+ ‘ 𝑅 ) 𝑀𝑀 ( ( t+ ‘ 𝑅 ) ∪ I ) 𝑌 ) ) ) )