Theorem ex-sategoelel12 35421

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 8447	. . . . . . . 8 ⊢ 1o ∈ V
3	2	prid1 4729	. . . . . . 7 ⊢ 1o ∈ {1o, 2o}
4		2oex 8448	. . . . . . . 8 ⊢ 2o ∈ V
5	4	prid2 4730	. . . . . . 7 ⊢ 2o ∈ {1o, 2o}
6	3, 5	ifcli 4539	. . . . . 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 7087	. . . 4 ⊢ 𝑆:ω⟶{1o, 2o}
9		prex 5395	. . . . 5 ⊢ {1o, 2o} ∈ V
10		omex 9603	. . . . 5 ⊢ ω ∈ V
11	9, 10	elmap 8847	. . . 4 ⊢ (𝑆 ∈ ({1o, 2o} ↑m ω) ↔ 𝑆:ω⟶{1o, 2o})
12	8, 11	mpbir 231	. . 3 ⊢ 𝑆 ∈ ({1o, 2o} ↑m ω)
13	2	sucid 6419	. . . . 5 ⊢ 1o ∈ suc 1o
14		df-2o 8438	. . . . 5 ⊢ 2o = suc 1o
15	13, 14	eleqtrri 2828	. . . 4 ⊢ 1o ∈ 2o
16		2onn 8609	. . . . 5 ⊢ 2o ∈ ω
17		1onn 8607	. . . . 5 ⊢ 1o ∈ ω
18		iftrue 4497	. . . . . 6 ⊢ (𝑥 = 2o → if(𝑥 = 2o, 1o, 2o) = 1o)
19	18, 1	fvmptg 6969	. . . . 5 ⊢ ((2o ∈ ω ∧ 1o ∈ ω) → (𝑆‘2o) = 1o)
20	16, 17, 19	mp2an 692	. . . 4 ⊢ (𝑆‘2o) = 1o
21		1one2o 8613	. . . . . . . . 9 ⊢ 1o ≠ 2o
22	21	neii 2928	. . . . . . . 8 ⊢ ¬ 1o = 2o
23		eqeq1 2734	. . . . . . . 8 ⊢ (𝑥 = 1o → (𝑥 = 2o ↔ 1o = 2o))
24	22, 23	mtbiri 327	. . . . . . 7 ⊢ (𝑥 = 1o → ¬ 𝑥 = 2o)
25	24	iffalsed 4502	. . . . . 6 ⊢ (𝑥 = 1o → if(𝑥 = 2o, 1o, 2o) = 2o)
26	25, 1	fvmptg 6969	. . . . 5 ⊢ ((1o ∈ ω ∧ 2o ∈ ω) → (𝑆‘1o) = 2o)
27	17, 16, 26	mp2an 692	. . . 4 ⊢ (𝑆‘1o) = 2o
28	15, 20, 27	3eltr4i 2842	. . 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 2730	. . . 4 ⊢ ({1o, 2o} Sat∈ (2o∈𝑔1o)) = ({1o, 2o} Sat∈ (2o∈𝑔1o))
32	31	sategoelfvb 35413	. . 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 3450  ifcif 4491  {cpr 4594   ↦ cmpt 5191  suc csuc 6337  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  ωcom 7845  1_oc1o 8430  2_oc2o 8431   ↑_m cmap 8802  ∈_𝑔cgoe 35327   Sat_∈ csate 35332

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-ac2 10423

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-card 9899  df-ac 10076  df-goel 35334  df-gona 35335  df-goal 35336  df-sat 35337  df-sate 35338  df-fmla 35339

This theorem is referenced by: (None)