Theorem ex-sategoelel12 35255

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 8506	. . . . . . . 8 ⊢ 1o ∈ V
3	2	prid1 4771	. . . . . . 7 ⊢ 1o ∈ {1o, 2o}
4		2oex 8507	. . . . . . . 8 ⊢ 2o ∈ V
5	4	prid2 4772	. . . . . . 7 ⊢ 2o ∈ {1o, 2o}
6	3, 5	ifcli 4580	. . . . . 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 7126	. . . 4 ⊢ 𝑆:ω⟶{1o, 2o}
9		prex 5438	. . . . 5 ⊢ {1o, 2o} ∈ V
10		omex 9686	. . . . 5 ⊢ ω ∈ V
11	9, 10	elmap 8900	. . . 4 ⊢ (𝑆 ∈ ({1o, 2o} ↑m ω) ↔ 𝑆:ω⟶{1o, 2o})
12	8, 11	mpbir 230	. . 3 ⊢ 𝑆 ∈ ({1o, 2o} ↑m ω)
13	2	sucid 6458	. . . . 5 ⊢ 1o ∈ suc 1o
14		df-2o 8497	. . . . 5 ⊢ 2o = suc 1o
15	13, 14	eleqtrri 2825	. . . 4 ⊢ 1o ∈ 2o
16		2onn 8672	. . . . 5 ⊢ 2o ∈ ω
17		1onn 8670	. . . . 5 ⊢ 1o ∈ ω
18		iftrue 4539	. . . . . 6 ⊢ (𝑥 = 2o → if(𝑥 = 2o, 1o, 2o) = 1o)
19	18, 1	fvmptg 7007	. . . . 5 ⊢ ((2o ∈ ω ∧ 1o ∈ ω) → (𝑆‘2o) = 1o)
20	16, 17, 19	mp2an 690	. . . 4 ⊢ (𝑆‘2o) = 1o
21		1one2o 8676	. . . . . . . . 9 ⊢ 1o ≠ 2o
22	21	neii 2932	. . . . . . . 8 ⊢ ¬ 1o = 2o
23		eqeq1 2730	. . . . . . . 8 ⊢ (𝑥 = 1o → (𝑥 = 2o ↔ 1o = 2o))
24	22, 23	mtbiri 326	. . . . . . 7 ⊢ (𝑥 = 1o → ¬ 𝑥 = 2o)
25	24	iffalsed 4544	. . . . . 6 ⊢ (𝑥 = 1o → if(𝑥 = 2o, 1o, 2o) = 2o)
26	25, 1	fvmptg 7007	. . . . 5 ⊢ ((1o ∈ ω ∧ 2o ∈ ω) → (𝑆‘1o) = 2o)
27	17, 16, 26	mp2an 690	. . . 4 ⊢ (𝑆‘1o) = 2o
28	15, 20, 27	3eltr4i 2839	. . 3 ⊢ (𝑆‘2o) ∈ (𝑆‘1o)
29	12, 28	pm3.2i 469	. 2 ⊢ (𝑆 ∈ ({1o, 2o} ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o))
30	16, 17	pm3.2i 469	. . 3 ⊢ (2o ∈ ω ∧ 1o ∈ ω)
31		eqid 2726	. . . 4 ⊢ ({1o, 2o} Sat∈ (2o∈𝑔1o)) = ({1o, 2o} Sat∈ (2o∈𝑔1o))
32	31	sategoelfvb 35247	. . 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 690	. 2 ⊢ (𝑆 ∈ ({1o, 2o} Sat∈ (2o∈𝑔1o)) ↔ (𝑆 ∈ ({1o, 2o} ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o)))
34	29, 33	mpbir 230	1 ⊢ 𝑆 ∈ ({1o, 2o} Sat∈ (2o∈𝑔1o))

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099  Vcvv 3462  ifcif 4533  {cpr 4635   ↦ cmpt 5236  suc csuc 6378  ⟶wf 6550  ‘cfv 6554  (class class class)co 7424  ωcom 7876  1_oc1o 8489  2_oc2o 8490   ↑_m cmap 8855  ∈_𝑔cgoe 35161   Sat_∈ csate 35166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9684  ax-ac2 10506

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-er 8734  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-card 9982  df-ac 10159  df-goel 35168  df-gona 35169  df-goal 35170  df-sat 35171  df-sate 35172  df-fmla 35173

This theorem is referenced by: (None)