Theorem ex-sategoelelomsuc 35477

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: (𝑆‘2_o) = 𝑍 ∈ suc 𝑍 = (𝑆‘2_o). Remark: the indices 1_o and 2_o are intentionally reversed to distinguish them from elements of the model: (2_o∈_𝑔1_o) should not be confused with 2_o ∈ 1_o, which is false. (Contributed by AV, 19-Nov-2023.)

Hypothesis

Ref	Expression
ex-sategoelelomsuc.s	⊢ 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 𝑍, suc 𝑍))

Assertion

Ref	Expression
ex-sategoelelomsuc	⊢ (𝑍 ∈ ω → 𝑆 ∈ (ω Sat∈ (2o∈𝑔1o)))

Distinct variable group:   𝑥,𝑍

Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem ex-sategoelelomsuc

Step	Hyp	Ref	Expression
1		id 22	. . . . . 6 ⊢ (𝑍 ∈ ω → 𝑍 ∈ ω)
2		peano2 7826	. . . . . 6 ⊢ (𝑍 ∈ ω → suc 𝑍 ∈ ω)
3	1, 2	ifcld 4521	. . . . 5 ⊢ (𝑍 ∈ ω → if(𝑥 = 2o, 𝑍, suc 𝑍) ∈ ω)
4	3	adantr 480	. . . 4 ⊢ ((𝑍 ∈ ω ∧ 𝑥 ∈ ω) → if(𝑥 = 2o, 𝑍, suc 𝑍) ∈ ω)
5		ex-sategoelelomsuc.s	. . . 4 ⊢ 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 𝑍, suc 𝑍))
6	4, 5	fmptd 7053	. . 3 ⊢ (𝑍 ∈ ω → 𝑆:ω⟶ω)
7		omex 9539	. . . . 5 ⊢ ω ∈ V
8	7	a1i 11	. . . 4 ⊢ (𝑍 ∈ ω → ω ∈ V)
9	8, 8	elmapd 8770	. . 3 ⊢ (𝑍 ∈ ω → (𝑆 ∈ (ω ↑m ω) ↔ 𝑆:ω⟶ω))
10	6, 9	mpbird 257	. 2 ⊢ (𝑍 ∈ ω → 𝑆 ∈ (ω ↑m ω))
11		sucidg 6395	. . 3 ⊢ (𝑍 ∈ ω → 𝑍 ∈ suc 𝑍)
12	5	a1i 11	. . . 4 ⊢ (𝑍 ∈ ω → 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 𝑍, suc 𝑍)))
13		iftrue 4480	. . . . 5 ⊢ (𝑥 = 2o → if(𝑥 = 2o, 𝑍, suc 𝑍) = 𝑍)
14	13	adantl 481	. . . 4 ⊢ ((𝑍 ∈ ω ∧ 𝑥 = 2o) → if(𝑥 = 2o, 𝑍, suc 𝑍) = 𝑍)
15		2onn 8563	. . . . 5 ⊢ 2o ∈ ω
16	15	a1i 11	. . . 4 ⊢ (𝑍 ∈ ω → 2o ∈ ω)
17	12, 14, 16, 1	fvmptd 6942	. . 3 ⊢ (𝑍 ∈ ω → (𝑆‘2o) = 𝑍)
18		1one2o 8567	. . . . . . . 8 ⊢ 1o ≠ 2o
19	18	neii 2930	. . . . . . 7 ⊢ ¬ 1o = 2o
20		eqeq1 2735	. . . . . . 7 ⊢ (𝑥 = 1o → (𝑥 = 2o ↔ 1o = 2o))
21	19, 20	mtbiri 327	. . . . . 6 ⊢ (𝑥 = 1o → ¬ 𝑥 = 2o)
22	21	iffalsed 4485	. . . . 5 ⊢ (𝑥 = 1o → if(𝑥 = 2o, 𝑍, suc 𝑍) = suc 𝑍)
23	22	adantl 481	. . . 4 ⊢ ((𝑍 ∈ ω ∧ 𝑥 = 1o) → if(𝑥 = 2o, 𝑍, suc 𝑍) = suc 𝑍)
24		1onn 8561	. . . . 5 ⊢ 1o ∈ ω
25	24	a1i 11	. . . 4 ⊢ (𝑍 ∈ ω → 1o ∈ ω)
26	12, 23, 25, 2	fvmptd 6942	. . 3 ⊢ (𝑍 ∈ ω → (𝑆‘1o) = suc 𝑍)
27	11, 17, 26	3eltr4d 2846	. 2 ⊢ (𝑍 ∈ ω → (𝑆‘2o) ∈ (𝑆‘1o))
28	15, 24	pm3.2i 470	. . . 4 ⊢ (2o ∈ ω ∧ 1o ∈ ω)
29	7, 28	pm3.2i 470	. . 3 ⊢ (ω ∈ V ∧ (2o ∈ ω ∧ 1o ∈ ω))
30		eqid 2731	. . . 4 ⊢ (ω Sat∈ (2o∈𝑔1o)) = (ω Sat∈ (2o∈𝑔1o))
31	30	sategoelfvb 35470	. . 3 ⊢ ((ω ∈ V ∧ (2o ∈ ω ∧ 1o ∈ ω)) → (𝑆 ∈ (ω Sat∈ (2o∈𝑔1o)) ↔ (𝑆 ∈ (ω ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o))))
32	29, 31	mp1i 13	. 2 ⊢ (𝑍 ∈ ω → (𝑆 ∈ (ω Sat∈ (2o∈𝑔1o)) ↔ (𝑆 ∈ (ω ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o))))
33	10, 27, 32	mpbir2and 713	1 ⊢ (𝑍 ∈ ω → 𝑆 ∈ (ω Sat∈ (2o∈𝑔1o)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ifcif 4474   ↦ cmpt 5174  suc csuc 6314  ⟶wf 6483  ‘cfv 6487  (class class class)co 7352  ωcom 7802  1_oc1o 8384  2_oc2o 8385   ↑_m cmap 8756  ∈_𝑔cgoe 35384   Sat_∈ csate 35389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-inf2 9537  ax-ac2 10360

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-isom 6496  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-er 8628  df-map 8758  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-card 9838  df-ac 10013  df-goel 35391  df-gona 35392  df-goal 35393  df-sat 35394  df-sate 35395  df-fmla 35396

This theorem is referenced by: (None)