Metamath Proof Explorer


Theorem ex-sategoelelomsuc

Description: Example of a valuation of a simplified satisfaction predicate over the ordinal numbers as model for a Godel-set of membership using the properties of a successor: ( S2o ) = Z e. suc Z = ( S2o ) . Remark: the indices 1o and 2o are intentionally reversed to distinguish them from elements of the model: ( 2o e.g 1o ) should not be confused with 2o e. 1o , which is false. (Contributed by AV, 19-Nov-2023)

Ref Expression
Hypothesis ex-sategoelelomsuc.s S = x ω if x = 2 𝑜 Z suc Z
Assertion ex-sategoelelomsuc Z ω S ω Sat 2 𝑜 𝑔 1 𝑜

Proof

Step Hyp Ref Expression
1 ex-sategoelelomsuc.s S = x ω if x = 2 𝑜 Z suc Z
2 id Z ω Z ω
3 peano2 Z ω suc Z ω
4 2 3 ifcld Z ω if x = 2 𝑜 Z suc Z ω
5 4 adantr Z ω x ω if x = 2 𝑜 Z suc Z ω
6 5 1 fmptd Z ω S : ω ω
7 omex ω V
8 7 a1i Z ω ω V
9 8 8 elmapd Z ω S ω ω S : ω ω
10 6 9 mpbird Z ω S ω ω
11 sucidg Z ω Z suc Z
12 1 a1i Z ω S = x ω if x = 2 𝑜 Z suc Z
13 iftrue x = 2 𝑜 if x = 2 𝑜 Z suc Z = Z
14 13 adantl Z ω x = 2 𝑜 if x = 2 𝑜 Z suc Z = Z
15 2onn 2 𝑜 ω
16 15 a1i Z ω 2 𝑜 ω
17 12 14 16 2 fvmptd Z ω S 2 𝑜 = Z
18 1one2o 1 𝑜 2 𝑜
19 18 neii ¬ 1 𝑜 = 2 𝑜
20 eqeq1 x = 1 𝑜 x = 2 𝑜 1 𝑜 = 2 𝑜
21 19 20 mtbiri x = 1 𝑜 ¬ x = 2 𝑜
22 21 iffalsed x = 1 𝑜 if x = 2 𝑜 Z suc Z = suc Z
23 22 adantl Z ω x = 1 𝑜 if x = 2 𝑜 Z suc Z = suc Z
24 1onn 1 𝑜 ω
25 24 a1i Z ω 1 𝑜 ω
26 12 23 25 3 fvmptd Z ω S 1 𝑜 = suc Z
27 11 17 26 3eltr4d Z ω S 2 𝑜 S 1 𝑜
28 15 24 pm3.2i 2 𝑜 ω 1 𝑜 ω
29 7 28 pm3.2i ω V 2 𝑜 ω 1 𝑜 ω
30 eqid ω Sat 2 𝑜 𝑔 1 𝑜 = ω Sat 2 𝑜 𝑔 1 𝑜
31 30 sategoelfvb ω V 2 𝑜 ω 1 𝑜 ω S ω Sat 2 𝑜 𝑔 1 𝑜 S ω ω S 2 𝑜 S 1 𝑜
32 29 31 mp1i Z ω S ω Sat 2 𝑜 𝑔 1 𝑜 S ω ω S 2 𝑜 S 1 𝑜
33 10 27 32 mpbir2and Z ω S ω Sat 2 𝑜 𝑔 1 𝑜