Theorem ex-sategoelel12 35407

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 8421	. . . . . . . 8 ⊢ 1o ∈ V
3	2	prid1 4722	. . . . . . 7 ⊢ 1o ∈ {1o, 2o}
4		2oex 8422	. . . . . . . 8 ⊢ 2o ∈ V
5	4	prid2 4723	. . . . . . 7 ⊢ 2o ∈ {1o, 2o}
6	3, 5	ifcli 4532	. . . . . 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 7066	. . . 4 ⊢ 𝑆:ω⟶{1o, 2o}
9		prex 5387	. . . . 5 ⊢ {1o, 2o} ∈ V
10		omex 9572	. . . . 5 ⊢ ω ∈ V
11	9, 10	elmap 8821	. . . 4 ⊢ (𝑆 ∈ ({1o, 2o} ↑m ω) ↔ 𝑆:ω⟶{1o, 2o})
12	8, 11	mpbir 231	. . 3 ⊢ 𝑆 ∈ ({1o, 2o} ↑m ω)
13	2	sucid 6404	. . . . 5 ⊢ 1o ∈ suc 1o
14		df-2o 8412	. . . . 5 ⊢ 2o = suc 1o
15	13, 14	eleqtrri 2827	. . . 4 ⊢ 1o ∈ 2o
16		2onn 8583	. . . . 5 ⊢ 2o ∈ ω
17		1onn 8581	. . . . 5 ⊢ 1o ∈ ω
18		iftrue 4490	. . . . . 6 ⊢ (𝑥 = 2o → if(𝑥 = 2o, 1o, 2o) = 1o)
19	18, 1	fvmptg 6948	. . . . 5 ⊢ ((2o ∈ ω ∧ 1o ∈ ω) → (𝑆‘2o) = 1o)
20	16, 17, 19	mp2an 692	. . . 4 ⊢ (𝑆‘2o) = 1o
21		1one2o 8587	. . . . . . . . 9 ⊢ 1o ≠ 2o
22	21	neii 2927	. . . . . . . 8 ⊢ ¬ 1o = 2o
23		eqeq1 2733	. . . . . . . 8 ⊢ (𝑥 = 1o → (𝑥 = 2o ↔ 1o = 2o))
24	22, 23	mtbiri 327	. . . . . . 7 ⊢ (𝑥 = 1o → ¬ 𝑥 = 2o)
25	24	iffalsed 4495	. . . . . 6 ⊢ (𝑥 = 1o → if(𝑥 = 2o, 1o, 2o) = 2o)
26	25, 1	fvmptg 6948	. . . . 5 ⊢ ((1o ∈ ω ∧ 2o ∈ ω) → (𝑆‘1o) = 2o)
27	17, 16, 26	mp2an 692	. . . 4 ⊢ (𝑆‘1o) = 2o
28	15, 20, 27	3eltr4i 2841	. . 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 2729	. . . 4 ⊢ ({1o, 2o} Sat∈ (2o∈𝑔1o)) = ({1o, 2o} Sat∈ (2o∈𝑔1o))
32	31	sategoelfvb 35399	. . 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 2109  Vcvv 3444  ifcif 4484  {cpr 4587   ↦ cmpt 5183  suc csuc 6322  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  ωcom 7822  1_oc1o 8404  2_oc2o 8405   ↑_m cmap 8776  ∈_𝑔cgoe 35313   Sat_∈ csate 35318

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-inf2 9570  ax-ac2 10392

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 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-card 9868  df-ac 10045  df-goel 35320  df-gona 35321  df-goal 35322  df-sat 35323  df-sate 35324  df-fmla 35325

This theorem is referenced by: (None)