Theorem ex-sategoelelomsuc 35992

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 23	. . . . . 6 ⊢ (𝑍 ∈ ω → 𝑍 ∈ ω)
2		peano2 7889	. . . . . 6 ⊢ (𝑍 ∈ ω → suc 𝑍 ∈ ω)
3	1, 2	ifcld 4532	. . . . 5 ⊢ (𝑍 ∈ ω → if(𝑥 = 2o, 𝑍, suc 𝑍) ∈ ω)
4	3	adantr 486	. . . 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 7110	. . 3 ⊢ (𝑍 ∈ ω → 𝑆:ω⟶ω)
7		omex 9625	. . . . 5 ⊢ ω ∈ V
8	7	a1i 11	. . . 4 ⊢ (𝑍 ∈ ω → ω ∈ V)
9	8, 8	elmapd 8842	. . 3 ⊢ (𝑍 ∈ ω → (𝑆 ∈ (ω ↑m ω) ↔ 𝑆:ω⟶ω))
10	6, 9	mpbird 260	. 2 ⊢ (𝑍 ∈ ω → 𝑆 ∈ (ω ↑m ω))
11		sucidg 6445	. . 3 ⊢ (𝑍 ∈ ω → 𝑍 ∈ suc 𝑍)
12	5	a1i 11	. . . 4 ⊢ (𝑍 ∈ ω → 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 𝑍, suc 𝑍)))
13		iftrue 4491	. . . . 5 ⊢ (𝑥 = 2o → if(𝑥 = 2o, 𝑍, suc 𝑍) = 𝑍)
14	13	adantl 487	. . . 4 ⊢ ((𝑍 ∈ ω ∧ 𝑥 = 2o) → if(𝑥 = 2o, 𝑍, suc 𝑍) = 𝑍)
15		2onn 8633	. . . . 5 ⊢ 2o ∈ ω
16	15	a1i 11	. . . 4 ⊢ (𝑍 ∈ ω → 2o ∈ ω)
17	12, 14, 16, 1	fvmptd 6998	. . 3 ⊢ (𝑍 ∈ ω → (𝑆‘2o) = 𝑍)
18		1one2o 8637	. . . . . . . 8 ⊢ 1o ≠ 2o
19	18	neii 2959	. . . . . . 7 ⊢ ¬ 1o = 2o
20		eqeq1 2766	. . . . . . 7 ⊢ (𝑥 = 1o → (𝑥 = 2o ↔ 1o = 2o))
21	19, 20	mtbiri 330	. . . . . 6 ⊢ (𝑥 = 1o → ¬ 𝑥 = 2o)
22	21	iffalsed 4496	. . . . 5 ⊢ (𝑥 = 1o → if(𝑥 = 2o, 𝑍, suc 𝑍) = suc 𝑍)
23	22	adantl 487	. . . 4 ⊢ ((𝑍 ∈ ω ∧ 𝑥 = 1o) → if(𝑥 = 2o, 𝑍, suc 𝑍) = suc 𝑍)
24		1onn 8631	. . . . 5 ⊢ 1o ∈ ω
25	24	a1i 11	. . . 4 ⊢ (𝑍 ∈ ω → 1o ∈ ω)
26	12, 23, 25, 2	fvmptd 6998	. . 3 ⊢ (𝑍 ∈ ω → (𝑆‘1o) = suc 𝑍)
27	11, 17, 26	3eltr4d 2877	. 2 ⊢ (𝑍 ∈ ω → (𝑆‘2o) ∈ (𝑆‘1o))
28	15, 24	pm3.2i 476	. . . 4 ⊢ (2o ∈ ω ∧ 1o ∈ ω)
29	7, 28	pm3.2i 476	. . 3 ⊢ (ω ∈ V ∧ (2o ∈ ω ∧ 1o ∈ ω))
30		eqid 2762	. . . 4 ⊢ (ω Sat∈ (2o∈𝑔1o)) = (ω Sat∈ (2o∈𝑔1o))
31	30	sategoelfvb 35985	. . 3 ⊢ ((ω ∈ V ∧ (2o ∈ ω ∧ 1o ∈ ω)) → (𝑆 ∈ (ω Sat∈ (2o∈𝑔1o)) ↔ (𝑆 ∈ (ω ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o))))
32	29, 31	mp1i 14	. 2 ⊢ (𝑍 ∈ ω → (𝑆 ∈ (ω Sat∈ (2o∈𝑔1o)) ↔ (𝑆 ∈ (ω ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o))))
33	10, 27, 32	mpbir2and 726	1 ⊢ (𝑍 ∈ ω → 𝑆 ∈ (ω Sat∈ (2o∈𝑔1o)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453  ifcif 4485   ↦ cmpt 5190  suc csuc 6363  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416  ωcom 7865  1_oc1o 8451  2_oc2o 8452   ↑_m cmap 8829  ∈_𝑔cgoe 35899   Sat_∈ csate 35904

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-inf2 9623  ax-ac2 10468

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-card 9947  df-ac 10122  df-goel 35906  df-gona 35907  df-goal 35908  df-sat 35909  df-sate 35910  df-fmla 35911

This theorem is used by: (None)