Theorem ex-sategoelelomsuc 35413

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 7866	. . . . . 6 ⊢ (𝑍 ∈ ω → suc 𝑍 ∈ ω)
3	1, 2	ifcld 4535	. . . . 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 7086	. . 3 ⊢ (𝑍 ∈ ω → 𝑆:ω⟶ω)
7		omex 9596	. . . . 5 ⊢ ω ∈ V
8	7	a1i 11	. . . 4 ⊢ (𝑍 ∈ ω → ω ∈ V)
9	8, 8	elmapd 8813	. . 3 ⊢ (𝑍 ∈ ω → (𝑆 ∈ (ω ↑m ω) ↔ 𝑆:ω⟶ω))
10	6, 9	mpbird 257	. 2 ⊢ (𝑍 ∈ ω → 𝑆 ∈ (ω ↑m ω))
11		sucidg 6415	. . 3 ⊢ (𝑍 ∈ ω → 𝑍 ∈ suc 𝑍)
12	5	a1i 11	. . . 4 ⊢ (𝑍 ∈ ω → 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 𝑍, suc 𝑍)))
13		iftrue 4494	. . . . 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 8606	. . . . 5 ⊢ 2o ∈ ω
16	15	a1i 11	. . . 4 ⊢ (𝑍 ∈ ω → 2o ∈ ω)
17	12, 14, 16, 1	fvmptd 6975	. . 3 ⊢ (𝑍 ∈ ω → (𝑆‘2o) = 𝑍)
18		1one2o 8610	. . . . . . . 8 ⊢ 1o ≠ 2o
19	18	neii 2927	. . . . . . 7 ⊢ ¬ 1o = 2o
20		eqeq1 2733	. . . . . . 7 ⊢ (𝑥 = 1o → (𝑥 = 2o ↔ 1o = 2o))
21	19, 20	mtbiri 327	. . . . . 6 ⊢ (𝑥 = 1o → ¬ 𝑥 = 2o)
22	21	iffalsed 4499	. . . . 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 8604	. . . . 5 ⊢ 1o ∈ ω
25	24	a1i 11	. . . 4 ⊢ (𝑍 ∈ ω → 1o ∈ ω)
26	12, 23, 25, 2	fvmptd 6975	. . 3 ⊢ (𝑍 ∈ ω → (𝑆‘1o) = suc 𝑍)
27	11, 17, 26	3eltr4d 2843	. 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 2729	. . . 4 ⊢ (ω Sat∈ (2o∈𝑔1o)) = (ω Sat∈ (2o∈𝑔1o))
31	30	sategoelfvb 35406	. . 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 1540   ∈ wcel 2109  Vcvv 3447  ifcif 4488   ↦ cmpt 5188  suc csuc 6334  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  ωcom 7842  1_oc1o 8427  2_oc2o 8428   ↑_m cmap 8799  ∈_𝑔cgoe 35320   Sat_∈ csate 35325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594  ax-ac2 10416

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-card 9892  df-ac 10069  df-goel 35327  df-gona 35328  df-goal 35329  df-sat 35330  df-sate 35331  df-fmla 35332

This theorem is referenced by: (None)