Theorem ex-sategoelelomsuc 33288

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

Assertion

Ref	Expression
ex-sategoelelomsuc	⊢ (𝑍 ∈ ω → 𝑆 ∈ (ω Sat∈ (2o∈𝑔1o)))

Distinct variable group:   𝑥,𝑍

Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem ex-sategoelelomsuc

Step	Hyp	Ref	Expression
1		id 22	. . . . . 6 ⊢ (𝑍 ∈ ω → 𝑍 ∈ ω)
2		peano2 7711	. . . . . 6 ⊢ (𝑍 ∈ ω → suc 𝑍 ∈ ω)
3	1, 2	ifcld 4502	. . . . 5 ⊢ (𝑍 ∈ ω → if(𝑥 = 2o, 𝑍, suc 𝑍) ∈ ω)
4	3	adantr 480	. . . 4 ⊢ ((𝑍 ∈ ω ∧ 𝑥 ∈ ω) → if(𝑥 = 2o, 𝑍, suc 𝑍) ∈ ω)
5		ex-sategoelelomsuc.s	. . . 4 ⊢ 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 𝑍, suc 𝑍))
6	4, 5	fmptd 6970	. . 3 ⊢ (𝑍 ∈ ω → 𝑆:ω⟶ω)
7		omex 9331	. . . . 5 ⊢ ω ∈ V
8	7	a1i 11	. . . 4 ⊢ (𝑍 ∈ ω → ω ∈ V)
9	8, 8	elmapd 8587	. . 3 ⊢ (𝑍 ∈ ω → (𝑆 ∈ (ω ↑m ω) ↔ 𝑆:ω⟶ω))
10	6, 9	mpbird 256	. 2 ⊢ (𝑍 ∈ ω → 𝑆 ∈ (ω ↑m ω))
11		sucidg 6329	. . 3 ⊢ (𝑍 ∈ ω → 𝑍 ∈ suc 𝑍)
12	5	a1i 11	. . . 4 ⊢ (𝑍 ∈ ω → 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 𝑍, suc 𝑍)))
13		iftrue 4462	. . . . 5 ⊢ (𝑥 = 2o → if(𝑥 = 2o, 𝑍, suc 𝑍) = 𝑍)
14	13	adantl 481	. . . 4 ⊢ ((𝑍 ∈ ω ∧ 𝑥 = 2o) → if(𝑥 = 2o, 𝑍, suc 𝑍) = 𝑍)
15		2onn 8433	. . . . 5 ⊢ 2o ∈ ω
16	15	a1i 11	. . . 4 ⊢ (𝑍 ∈ ω → 2o ∈ ω)
17	12, 14, 16, 1	fvmptd 6864	. . 3 ⊢ (𝑍 ∈ ω → (𝑆‘2o) = 𝑍)
18		1one2o 8436	. . . . . . . 8 ⊢ 1o ≠ 2o
19	18	neii 2944	. . . . . . 7 ⊢ ¬ 1o = 2o
20		eqeq1 2742	. . . . . . 7 ⊢ (𝑥 = 1o → (𝑥 = 2o ↔ 1o = 2o))
21	19, 20	mtbiri 326	. . . . . 6 ⊢ (𝑥 = 1o → ¬ 𝑥 = 2o)
22	21	iffalsed 4467	. . . . 5 ⊢ (𝑥 = 1o → if(𝑥 = 2o, 𝑍, suc 𝑍) = suc 𝑍)
23	22	adantl 481	. . . 4 ⊢ ((𝑍 ∈ ω ∧ 𝑥 = 1o) → if(𝑥 = 2o, 𝑍, suc 𝑍) = suc 𝑍)
24		1onn 8432	. . . . 5 ⊢ 1o ∈ ω
25	24	a1i 11	. . . 4 ⊢ (𝑍 ∈ ω → 1o ∈ ω)
26	12, 23, 25, 2	fvmptd 6864	. . 3 ⊢ (𝑍 ∈ ω → (𝑆‘1o) = suc 𝑍)
27	11, 17, 26	3eltr4d 2854	. 2 ⊢ (𝑍 ∈ ω → (𝑆‘2o) ∈ (𝑆‘1o))
28	15, 24	pm3.2i 470	. . . 4 ⊢ (2o ∈ ω ∧ 1o ∈ ω)
29	7, 28	pm3.2i 470	. . 3 ⊢ (ω ∈ V ∧ (2o ∈ ω ∧ 1o ∈ ω))
30		eqid 2738	. . . 4 ⊢ (ω Sat∈ (2o∈𝑔1o)) = (ω Sat∈ (2o∈𝑔1o))
31	30	sategoelfvb 33281	. . 3 ⊢ ((ω ∈ V ∧ (2o ∈ ω ∧ 1o ∈ ω)) → (𝑆 ∈ (ω Sat∈ (2o∈𝑔1o)) ↔ (𝑆 ∈ (ω ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o))))
32	29, 31	mp1i 13	. 2 ⊢ (𝑍 ∈ ω → (𝑆 ∈ (ω Sat∈ (2o∈𝑔1o)) ↔ (𝑆 ∈ (ω ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o))))
33	10, 27, 32	mpbir2and 709	1 ⊢ (𝑍 ∈ ω → 𝑆 ∈ (ω Sat∈ (2o∈𝑔1o)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ifcif 4456   ↦ cmpt 5153  suc csuc 6253  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  ωcom 7687  1_oc1o 8260  2_oc2o 8261   ↑_m cmap 8573  ∈_𝑔cgoe 33195   Sat_∈ csate 33200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-ac2 10150

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-card 9628  df-ac 9803  df-goel 33202  df-gona 33203  df-goal 33204  df-sat 33205  df-sate 33206  df-fmla 33207

This theorem is referenced by: (None)