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Theorem ex-sategoelel12 34078
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 8423 . . . . . . . 8 1o ∈ V
32prid1 4724 . . . . . . 7 1o ∈ {1o, 2o}
4 2oex 8424 . . . . . . . 8 2o ∈ V
54prid2 4725 . . . . . . 7 2o ∈ {1o, 2o}
63, 5ifcli 4534 . . . . . 6 if(𝑥 = 2o, 1o, 2o) ∈ {1o, 2o}
76a1i 11 . . . . 5 (𝑥 ∈ ω → if(𝑥 = 2o, 1o, 2o) ∈ {1o, 2o})
81, 7fmpti 7061 . . . 4 𝑆:ω⟶{1o, 2o}
9 prex 5390 . . . . 5 {1o, 2o} ∈ V
10 omex 9584 . . . . 5 ω ∈ V
119, 10elmap 8812 . . . 4 (𝑆 ∈ ({1o, 2o} ↑m ω) ↔ 𝑆:ω⟶{1o, 2o})
128, 11mpbir 230 . . 3 𝑆 ∈ ({1o, 2o} ↑m ω)
132sucid 6400 . . . . 5 1o ∈ suc 1o
14 df-2o 8414 . . . . 5 2o = suc 1o
1513, 14eleqtrri 2833 . . . 4 1o ∈ 2o
16 2onn 8589 . . . . 5 2o ∈ ω
17 1onn 8587 . . . . 5 1o ∈ ω
18 iftrue 4493 . . . . . 6 (𝑥 = 2o → if(𝑥 = 2o, 1o, 2o) = 1o)
1918, 1fvmptg 6947 . . . . 5 ((2o ∈ ω ∧ 1o ∈ ω) → (𝑆‘2o) = 1o)
2016, 17, 19mp2an 691 . . . 4 (𝑆‘2o) = 1o
21 1one2o 8593 . . . . . . . . 9 1o ≠ 2o
2221neii 2942 . . . . . . . 8 ¬ 1o = 2o
23 eqeq1 2737 . . . . . . . 8 (𝑥 = 1o → (𝑥 = 2o ↔ 1o = 2o))
2422, 23mtbiri 327 . . . . . . 7 (𝑥 = 1o → ¬ 𝑥 = 2o)
2524iffalsed 4498 . . . . . 6 (𝑥 = 1o → if(𝑥 = 2o, 1o, 2o) = 2o)
2625, 1fvmptg 6947 . . . . 5 ((1o ∈ ω ∧ 2o ∈ ω) → (𝑆‘1o) = 2o)
2717, 16, 26mp2an 691 . . . 4 (𝑆‘1o) = 2o
2815, 20, 273eltr4i 2847 . . 3 (𝑆‘2o) ∈ (𝑆‘1o)
2912, 28pm3.2i 472 . 2 (𝑆 ∈ ({1o, 2o} ↑m ω) ∧ (𝑆‘2o) ∈ (𝑆‘1o))
3016, 17pm3.2i 472 . . 3 (2o ∈ ω ∧ 1o ∈ ω)
31 eqid 2733 . . . 4 ({1o, 2o} Sat (2o𝑔1o)) = ({1o, 2o} Sat (2o𝑔1o))
3231sategoelfvb 34070 . . 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 230 1 𝑆 ∈ ({1o, 2o} Sat (2o𝑔1o))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3444  ifcif 4487  {cpr 4589  cmpt 5189  suc csuc 6320  wf 6493  cfv 6497  (class class class)co 7358  ωcom 7803  1oc1o 8406  2oc2o 8407  m cmap 8768  𝑔cgoe 33984   Sat csate 33989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582  ax-ac2 10404
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-card 9880  df-ac 10057  df-goel 33991  df-gona 33992  df-goal 33993  df-sat 33994  df-sate 33995  df-fmla 33996
This theorem is referenced by: (None)
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