Theorem ex-sategoelel12 35852

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 8463	. . . . . . . 8 ⊢ 1o ∈ V
3	2	prid1 4733	. . . . . . 7 ⊢ 1o ∈ {1o, 2o}
4		2oex 8465	. . . . . . . 8 ⊢ 2o ∈ V
5	4	prid2 4734	. . . . . . 7 ⊢ 2o ∈ {1o, 2o}
6	3, 5	ifcli 4540	. . . . . 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 7108	. . . 4 ⊢ 𝑆:ω⟶{1o, 2o}
9		prex 5410	. . . . 5 ⊢ {1o, 2o} ∈ V
10		omex 9612	. . . . 5 ⊢ ω ∈ V
11	9, 10	elmap 8869	. . . 4 ⊢ (𝑆 ∈ ({1o, 2o} ↑m ω) ↔ 𝑆:ω⟶{1o, 2o})
12	8, 11	mpbir 234	. . 3 ⊢ 𝑆 ∈ ({1o, 2o} ↑m ω)
13	2	sucid 6446	. . . . 5 ⊢ 1o ∈ suc 1o
14		df-2o 8454	. . . . 5 ⊢ 2o = suc 1o
15	13, 14	eleqtrri 2868	. . . 4 ⊢ 1o ∈ 2o
16		2onn 8628	. . . . 5 ⊢ 2o ∈ ω
17		1onn 8626	. . . . 5 ⊢ 1o ∈ ω
18		iftrue 4498	. . . . . 6 ⊢ (𝑥 = 2o → if(𝑥 = 2o, 1o, 2o) = 1o)
19	18, 1	fvmptg 6988	. . . . 5 ⊢ ((2o ∈ ω ∧ 1o ∈ ω) → (𝑆‘2o) = 1o)
20	16, 17, 19	mp2an 704	. . . 4 ⊢ (𝑆‘2o) = 1o
21		1one2o 8632	. . . . . . . . 9 ⊢ 1o ≠ 2o
22	21	neii 2966	. . . . . . . 8 ⊢ ¬ 1o = 2o
23		eqeq1 2773	. . . . . . . 8 ⊢ (𝑥 = 1o → (𝑥 = 2o ↔ 1o = 2o))
24	22, 23	mtbiri 330	. . . . . . 7 ⊢ (𝑥 = 1o → ¬ 𝑥 = 2o)
25	24	iffalsed 4503	. . . . . 6 ⊢ (𝑥 = 1o → if(𝑥 = 2o, 1o, 2o) = 2o)
26	25, 1	fvmptg 6988	. . . . 5 ⊢ ((1o ∈ ω ∧ 2o ∈ ω) → (𝑆‘1o) = 2o)
27	17, 16, 26	mp2an 704	. . . 4 ⊢ (𝑆‘1o) = 2o
28	15, 20, 27	3eltr4i 2882	. . 3 ⊢ (𝑆‘2o) ∈ (𝑆‘1o)
29	12, 28	pm3.2i 475	. 2 ⊢ (𝑆 ∈ ({1o, 2o} ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o))
30	16, 17	pm3.2i 475	. . 3 ⊢ (2o ∈ ω ∧ 1o ∈ ω)
31		eqid 2769	. . . 4 ⊢ ({1o, 2o} Sat∈ (2o∈𝑔1o)) = ({1o, 2o} Sat∈ (2o∈𝑔1o))
32	31	sategoelfvb 35844	. . 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 704	. 2 ⊢ (𝑆 ∈ ({1o, 2o} Sat∈ (2o∈𝑔1o)) ↔ (𝑆 ∈ ({1o, 2o} ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o)))
34	29, 33	mpbir 234	1 ⊢ 𝑆 ∈ ({1o, 2o} Sat∈ (2o∈𝑔1o))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ifcif 4492  {cpr 4596   ↦ cmpt 5196  suc csuc 6363  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  ωcom 7862  1_oc1o 8446  2_oc2o 8447   ↑_m cmap 8824  ∈_𝑔cgoe 35758   Sat_∈ csate 35763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9610  ax-ac2 10447

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  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 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-er 8694  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-card 9925  df-ac 10100  df-goel 35765  df-gona 35766  df-goal 35767  df-sat 35768  df-sate 35769  df-fmla 35770

This theorem is referenced by: (None)