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Theorem ex-sategoelel12 35432
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: (2o𝑔1o) 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 8516 . . . . . . . 8 1o ∈ V
32prid1 4762 . . . . . . 7 1o ∈ {1o, 2o}
4 2oex 8517 . . . . . . . 8 2o ∈ V
54prid2 4763 . . . . . . 7 2o ∈ {1o, 2o}
63, 5ifcli 4573 . . . . . 6 if(𝑥 = 2o, 1o, 2o) ∈ {1o, 2o}
76a1i 11 . . . . 5 (𝑥 ∈ ω → if(𝑥 = 2o, 1o, 2o) ∈ {1o, 2o})
81, 7fmpti 7132 . . . 4 𝑆:ω⟶{1o, 2o}
9 prex 5437 . . . . 5 {1o, 2o} ∈ V
10 omex 9683 . . . . 5 ω ∈ V
119, 10elmap 8911 . . . 4 (𝑆 ∈ ({1o, 2o} ↑m ω) ↔ 𝑆:ω⟶{1o, 2o})
128, 11mpbir 231 . . 3 𝑆 ∈ ({1o, 2o} ↑m ω)
132sucid 6466 . . . . 5 1o ∈ suc 1o
14 df-2o 8507 . . . . 5 2o = suc 1o
1513, 14eleqtrri 2840 . . . 4 1o ∈ 2o
16 2onn 8680 . . . . 5 2o ∈ ω
17 1onn 8678 . . . . 5 1o ∈ ω
18 iftrue 4531 . . . . . 6 (𝑥 = 2o → if(𝑥 = 2o, 1o, 2o) = 1o)
1918, 1fvmptg 7014 . . . . 5 ((2o ∈ ω ∧ 1o ∈ ω) → (𝑆‘2o) = 1o)
2016, 17, 19mp2an 692 . . . 4 (𝑆‘2o) = 1o
21 1one2o 8684 . . . . . . . . 9 1o ≠ 2o
2221neii 2942 . . . . . . . 8 ¬ 1o = 2o
23 eqeq1 2741 . . . . . . . 8 (𝑥 = 1o → (𝑥 = 2o ↔ 1o = 2o))
2422, 23mtbiri 327 . . . . . . 7 (𝑥 = 1o → ¬ 𝑥 = 2o)
2524iffalsed 4536 . . . . . 6 (𝑥 = 1o → if(𝑥 = 2o, 1o, 2o) = 2o)
2625, 1fvmptg 7014 . . . . 5 ((1o ∈ ω ∧ 2o ∈ ω) → (𝑆‘1o) = 2o)
2717, 16, 26mp2an 692 . . . 4 (𝑆‘1o) = 2o
2815, 20, 273eltr4i 2854 . . 3 (𝑆‘2o) ∈ (𝑆‘1o)
2912, 28pm3.2i 470 . 2 (𝑆 ∈ ({1o, 2o} ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o))
3016, 17pm3.2i 470 . . 3 (2o ∈ ω ∧ 1o ∈ ω)
31 eqid 2737 . . . 4 ({1o, 2o} Sat (2o𝑔1o)) = ({1o, 2o} Sat (2o𝑔1o))
3231sategoelfvb 35424 . . 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 692 . 2 (𝑆 ∈ ({1o, 2o} Sat (2o𝑔1o)) ↔ (𝑆 ∈ ({1o, 2o} ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o)))
3429, 33mpbir 231 1 𝑆 ∈ ({1o, 2o} Sat (2o𝑔1o))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  ifcif 4525  {cpr 4628  cmpt 5225  suc csuc 6386  wf 6557  cfv 6561  (class class class)co 7431  ωcom 7887  1oc1o 8499  2oc2o 8500  m cmap 8866  𝑔cgoe 35338   Sat csate 35343
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-ac2 10503
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-ac 10156  df-goel 35345  df-gona 35346  df-goal 35347  df-sat 35348  df-sate 35349  df-fmla 35350
This theorem is referenced by: (None)
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