Users' Mathboxes 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 35412
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 8515 . . . . . . . 8 1o ∈ V
32prid1 4767 . . . . . . 7 1o ∈ {1o, 2o}
4 2oex 8516 . . . . . . . 8 2o ∈ V
54prid2 4768 . . . . . . 7 2o ∈ {1o, 2o}
63, 5ifcli 4578 . . . . . 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 5443 . . . . 5 {1o, 2o} ∈ V
10 omex 9681 . . . . 5 ω ∈ V
119, 10elmap 8910 . . . 4 (𝑆 ∈ ({1o, 2o} ↑m ω) ↔ 𝑆:ω⟶{1o, 2o})
128, 11mpbir 231 . . 3 𝑆 ∈ ({1o, 2o} ↑m ω)
132sucid 6468 . . . . 5 1o ∈ suc 1o
14 df-2o 8506 . . . . 5 2o = suc 1o
1513, 14eleqtrri 2838 . . . 4 1o ∈ 2o
16 2onn 8679 . . . . 5 2o ∈ ω
17 1onn 8677 . . . . 5 1o ∈ ω
18 iftrue 4537 . . . . . 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 8683 . . . . . . . . 9 1o ≠ 2o
2221neii 2940 . . . . . . . 8 ¬ 1o = 2o
23 eqeq1 2739 . . . . . . . 8 (𝑥 = 1o → (𝑥 = 2o ↔ 1o = 2o))
2422, 23mtbiri 327 . . . . . . 7 (𝑥 = 1o → ¬ 𝑥 = 2o)
2524iffalsed 4542 . . . . . 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 2852 . . 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 2735 . . . 4 ({1o, 2o} Sat (2o𝑔1o)) = ({1o, 2o} Sat (2o𝑔1o))
3231sategoelfvb 35404 . . 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 1537  wcel 2106  Vcvv 3478  ifcif 4531  {cpr 4633  cmpt 5231  suc csuc 6388  wf 6559  cfv 6563  (class class class)co 7431  ωcom 7887  1oc1o 8498  2oc2o 8499  m cmap 8865  𝑔cgoe 35318   Sat csate 35323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-ac2 10501
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-ac 10154  df-goel 35325  df-gona 35326  df-goal 35327  df-sat 35328  df-sate 35329  df-fmla 35330
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator