Mathbox for Mario Carneiro < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ex-sategoelel12 Structured version   Visualization version   GIF version

Theorem ex-sategoelel12 32853
 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‘2o) = 1o ∈ 2o = (𝑆‘2o). Remark: the indices 1o and 2o are intentionally reversed to distinguish them from elements of the model: (2o∈𝑔1o) should not be confused with 2o ∈ 1o, 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 (2o𝑔1o))

Proof of Theorem ex-sategoelel12
StepHypRef Expression
1 ex-sategoelel12.s . . . . 5 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 1o, 2o))
2 1oex 8111 . . . . . . . 8 1o ∈ V
32prid1 4661 . . . . . . 7 1o ∈ {1o, 2o}
4 2oex 8113 . . . . . . . 8 2o ∈ V
54prid2 4662 . . . . . . 7 2o ∈ {1o, 2o}
63, 5ifcli 4474 . . . . . 6 if(𝑥 = 2o, 1o, 2o) ∈ {1o, 2o}
76a1i 11 . . . . 5 (𝑥 ∈ ω → if(𝑥 = 2o, 1o, 2o) ∈ {1o, 2o})
81, 7fmpti 6863 . . . 4 𝑆:ω⟶{1o, 2o}
9 prex 5302 . . . . 5 {1o, 2o} ∈ V
10 omex 9108 . . . . 5 ω ∈ V
119, 10elmap 8436 . . . 4 (𝑆 ∈ ({1o, 2o} ↑m ω) ↔ 𝑆:ω⟶{1o, 2o})
128, 11mpbir 234 . . 3 𝑆 ∈ ({1o, 2o} ↑m ω)
132sucid 6245 . . . . 5 1o ∈ suc 1o
14 df-2o 8104 . . . . 5 2o = suc 1o
1513, 14eleqtrri 2889 . . . 4 1o ∈ 2o
16 2onn 8267 . . . . 5 2o ∈ ω
17 1onn 8266 . . . . 5 1o ∈ ω
18 iftrue 4434 . . . . . 6 (𝑥 = 2o → if(𝑥 = 2o, 1o, 2o) = 1o)
1918, 1fvmptg 6753 . . . . 5 ((2o ∈ ω ∧ 1o ∈ ω) → (𝑆‘2o) = 1o)
2016, 17, 19mp2an 691 . . . 4 (𝑆‘2o) = 1o
21 1one2o 8270 . . . . . . . . 9 1o ≠ 2o
2221neii 2989 . . . . . . . 8 ¬ 1o = 2o
23 eqeq1 2802 . . . . . . . 8 (𝑥 = 1o → (𝑥 = 2o ↔ 1o = 2o))
2422, 23mtbiri 330 . . . . . . 7 (𝑥 = 1o → ¬ 𝑥 = 2o)
2524iffalsed 4439 . . . . . 6 (𝑥 = 1o → if(𝑥 = 2o, 1o, 2o) = 2o)
2625, 1fvmptg 6753 . . . . 5 ((1o ∈ ω ∧ 2o ∈ ω) → (𝑆‘1o) = 2o)
2717, 16, 26mp2an 691 . . . 4 (𝑆‘1o) = 2o
2815, 20, 273eltr4i 2903 . . 3 (𝑆‘2o) ∈ (𝑆‘1o)
2912, 28pm3.2i 474 . 2 (𝑆 ∈ ({1o, 2o} ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o))
3016, 17pm3.2i 474 . . 3 (2o ∈ ω ∧ 1o ∈ ω)
31 eqid 2798 . . . 4 ({1o, 2o} Sat (2o𝑔1o)) = ({1o, 2o} Sat (2o𝑔1o))
3231sategoelfvb 32845 . . 3 (({1o, 2o} ∈ V ∧ (2o ∈ ω ∧ 1o ∈ ω)) → (𝑆 ∈ ({1o, 2o} Sat (2o𝑔1o)) ↔ (𝑆 ∈ ({1o, 2o} ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o))))
339, 30, 32mp2an 691 . 2 (𝑆 ∈ ({1o, 2o} Sat (2o𝑔1o)) ↔ (𝑆 ∈ ({1o, 2o} ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o)))
3429, 33mpbir 234 1 𝑆 ∈ ({1o, 2o} Sat (2o𝑔1o))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3442  ifcif 4428  {cpr 4530   ↦ cmpt 5114  suc csuc 6168  ⟶wf 6328  ‘cfv 6332  (class class class)co 7145  ωcom 7573  1oc1o 8096  2oc2o 8097   ↑m cmap 8407  ∈𝑔cgoe 32759   Sat∈ csate 32764 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-inf2 9106  ax-ac2 9892 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-isom 6341  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-2o 8104  df-er 8290  df-map 8409  df-en 8511  df-dom 8512  df-sdom 8513  df-card 9370  df-ac 9545  df-goel 32766  df-gona 32767  df-goal 32768  df-sat 32769  df-sate 32770  df-fmla 32771 This theorem is referenced by: (None)
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