Theorem ex-sategoelel12 35432

Description: Example of a valuation of a simplified satisfaction predicate over a proper pair (of ordinal numbers) as model for a Godel-set of membership using the properties of a successor: (𝑆‘2_o) = 1_o ∈ 2_o = (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-sategoelel12.s	⊢ 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 1o, 2o))

Assertion

Ref	Expression
ex-sategoelel12	⊢ 𝑆 ∈ ({1o, 2o} Sat∈ (2o∈𝑔1o))

Proof of Theorem ex-sategoelel12

Step	Hyp	Ref	Expression
1		ex-sategoelel12.s	. . . . 5 ⊢ 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 1o, 2o))
2		1oex 8516	. . . . . . . 8 ⊢ 1o ∈ V
3	2	prid1 4762	. . . . . . 7 ⊢ 1o ∈ {1o, 2o}
4		2oex 8517	. . . . . . . 8 ⊢ 2o ∈ V
5	4	prid2 4763	. . . . . . 7 ⊢ 2o ∈ {1o, 2o}
6	3, 5	ifcli 4573	. . . . . 6 ⊢ if(𝑥 = 2o, 1o, 2o) ∈ {1o, 2o}
7	6	a1i 11	. . . . 5 ⊢ (𝑥 ∈ ω → if(𝑥 = 2o, 1o, 2o) ∈ {1o, 2o})
8	1, 7	fmpti 7132	. . . 4 ⊢ 𝑆:ω⟶{1o, 2o}
9		prex 5437	. . . . 5 ⊢ {1o, 2o} ∈ V
10		omex 9683	. . . . 5 ⊢ ω ∈ V
11	9, 10	elmap 8911	. . . 4 ⊢ (𝑆 ∈ ({1o, 2o} ↑m ω) ↔ 𝑆:ω⟶{1o, 2o})
12	8, 11	mpbir 231	. . 3 ⊢ 𝑆 ∈ ({1o, 2o} ↑m ω)
13	2	sucid 6466	. . . . 5 ⊢ 1o ∈ suc 1o
14		df-2o 8507	. . . . 5 ⊢ 2o = suc 1o
15	13, 14	eleqtrri 2840	. . . 4 ⊢ 1o ∈ 2o
16		2onn 8680	. . . . 5 ⊢ 2o ∈ ω
17		1onn 8678	. . . . 5 ⊢ 1o ∈ ω
18		iftrue 4531	. . . . . 6 ⊢ (𝑥 = 2o → if(𝑥 = 2o, 1o, 2o) = 1o)
19	18, 1	fvmptg 7014	. . . . 5 ⊢ ((2o ∈ ω ∧ 1o ∈ ω) → (𝑆‘2o) = 1o)
20	16, 17, 19	mp2an 692	. . . 4 ⊢ (𝑆‘2o) = 1o
21		1one2o 8684	. . . . . . . . 9 ⊢ 1o ≠ 2o
22	21	neii 2942	. . . . . . . 8 ⊢ ¬ 1o = 2o
23		eqeq1 2741	. . . . . . . 8 ⊢ (𝑥 = 1o → (𝑥 = 2o ↔ 1o = 2o))
24	22, 23	mtbiri 327	. . . . . . 7 ⊢ (𝑥 = 1o → ¬ 𝑥 = 2o)
25	24	iffalsed 4536	. . . . . 6 ⊢ (𝑥 = 1o → if(𝑥 = 2o, 1o, 2o) = 2o)
26	25, 1	fvmptg 7014	. . . . 5 ⊢ ((1o ∈ ω ∧ 2o ∈ ω) → (𝑆‘1o) = 2o)
27	17, 16, 26	mp2an 692	. . . 4 ⊢ (𝑆‘1o) = 2o
28	15, 20, 27	3eltr4i 2854	. . 3 ⊢ (𝑆‘2o) ∈ (𝑆‘1o)
29	12, 28	pm3.2i 470	. 2 ⊢ (𝑆 ∈ ({1o, 2o} ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o))
30	16, 17	pm3.2i 470	. . 3 ⊢ (2o ∈ ω ∧ 1o ∈ ω)
31		eqid 2737	. . . 4 ⊢ ({1o, 2o} Sat∈ (2o∈𝑔1o)) = ({1o, 2o} Sat∈ (2o∈𝑔1o))
32	31	sategoelfvb 35424	. . 3 ⊢ (({1o, 2o} ∈ V ∧ (2o ∈ ω ∧ 1o ∈ ω)) → (𝑆 ∈ ({1o, 2o} Sat∈ (2o∈𝑔1o)) ↔ (𝑆 ∈ ({1o, 2o} ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o))))
33	9, 30, 32	mp2an 692	. 2 ⊢ (𝑆 ∈ ({1o, 2o} Sat∈ (2o∈𝑔1o)) ↔ (𝑆 ∈ ({1o, 2o} ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o)))
34	29, 33	mpbir 231	1 ⊢ 𝑆 ∈ ({1o, 2o} Sat∈ (2o∈𝑔1o))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ifcif 4525  {cpr 4628   ↦ 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)