Theorem ex-sategoelelomsuc 35431

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 7912	. . . . . 6 ⊢ (𝑍 ∈ ω → suc 𝑍 ∈ ω)
3	1, 2	ifcld 4572	. . . . 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 7134	. . 3 ⊢ (𝑍 ∈ ω → 𝑆:ω⟶ω)
7		omex 9683	. . . . 5 ⊢ ω ∈ V
8	7	a1i 11	. . . 4 ⊢ (𝑍 ∈ ω → ω ∈ V)
9	8, 8	elmapd 8880	. . 3 ⊢ (𝑍 ∈ ω → (𝑆 ∈ (ω ↑m ω) ↔ 𝑆:ω⟶ω))
10	6, 9	mpbird 257	. 2 ⊢ (𝑍 ∈ ω → 𝑆 ∈ (ω ↑m ω))
11		sucidg 6465	. . 3 ⊢ (𝑍 ∈ ω → 𝑍 ∈ suc 𝑍)
12	5	a1i 11	. . . 4 ⊢ (𝑍 ∈ ω → 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 𝑍, suc 𝑍)))
13		iftrue 4531	. . . . 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 8680	. . . . 5 ⊢ 2o ∈ ω
16	15	a1i 11	. . . 4 ⊢ (𝑍 ∈ ω → 2o ∈ ω)
17	12, 14, 16, 1	fvmptd 7023	. . 3 ⊢ (𝑍 ∈ ω → (𝑆‘2o) = 𝑍)
18		1one2o 8684	. . . . . . . 8 ⊢ 1o ≠ 2o
19	18	neii 2942	. . . . . . 7 ⊢ ¬ 1o = 2o
20		eqeq1 2741	. . . . . . 7 ⊢ (𝑥 = 1o → (𝑥 = 2o ↔ 1o = 2o))
21	19, 20	mtbiri 327	. . . . . 6 ⊢ (𝑥 = 1o → ¬ 𝑥 = 2o)
22	21	iffalsed 4536	. . . . 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 8678	. . . . 5 ⊢ 1o ∈ ω
25	24	a1i 11	. . . 4 ⊢ (𝑍 ∈ ω → 1o ∈ ω)
26	12, 23, 25, 2	fvmptd 7023	. . 3 ⊢ (𝑍 ∈ ω → (𝑆‘1o) = suc 𝑍)
27	11, 17, 26	3eltr4d 2856	. 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 2737	. . . 4 ⊢ (ω Sat∈ (2o∈𝑔1o)) = (ω Sat∈ (2o∈𝑔1o))
31	30	sategoelfvb 35424	. . 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 2108  Vcvv 3480  ifcif 4525   ↦ cmpt 5225  suc csuc 6386  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  ωcom 7887  1_oc1o 8499  2_oc2o 8500   ↑_m cmap 8866  ∈_𝑔cgoe 35338   Sat_∈ csate 35343

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-ac2 10503

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-ac 10156  df-goel 35345  df-gona 35346  df-goal 35347  df-sat 35348  df-sate 35349  df-fmla 35350

This theorem is referenced by: (None)