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Mirrors  >  Home  >  ILE Home  >  Th. List  >  unfiexmid GIF version

Theorem unfiexmid 6813
Description: If the union of any two finite sets is finite, excluded middle follows. Remark 8.1.17 of [AczelRathjen], p. 74. (Contributed by Mario Carneiro and Jim Kingdon, 5-Mar-2022.)
Hypothesis
Ref Expression
unfiexmid.1 ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥𝑦) ∈ Fin)
Assertion
Ref Expression
unfiexmid (𝜑 ∨ ¬ 𝜑)
Distinct variable group:   𝜑,𝑥,𝑦

Proof of Theorem unfiexmid
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-pr 3538 . . . . 5 {{𝑧 ∈ 1o𝜑}, 1o} = ({{𝑧 ∈ 1o𝜑}} ∪ {1o})
2 unfiexmid.1 . . . . . . 7 ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥𝑦) ∈ Fin)
32rgen2a 2489 . . . . . 6 𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥𝑦) ∈ Fin
4 df1o2 6333 . . . . . . . . . 10 1o = {∅}
5 rabeq 2681 . . . . . . . . . 10 (1o = {∅} → {𝑧 ∈ 1o𝜑} = {𝑧 ∈ {∅} ∣ 𝜑})
64, 5ax-mp 5 . . . . . . . . 9 {𝑧 ∈ 1o𝜑} = {𝑧 ∈ {∅} ∣ 𝜑}
7 ordtriexmidlem 4442 . . . . . . . . 9 {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
86, 7eqeltri 2213 . . . . . . . 8 {𝑧 ∈ 1o𝜑} ∈ On
9 snfig 6715 . . . . . . . 8 ({𝑧 ∈ 1o𝜑} ∈ On → {{𝑧 ∈ 1o𝜑}} ∈ Fin)
108, 9ax-mp 5 . . . . . . 7 {{𝑧 ∈ 1o𝜑}} ∈ Fin
11 1onn 6423 . . . . . . . 8 1o ∈ ω
12 snfig 6715 . . . . . . . 8 (1o ∈ ω → {1o} ∈ Fin)
1311, 12ax-mp 5 . . . . . . 7 {1o} ∈ Fin
14 uneq1 3227 . . . . . . . . 9 (𝑥 = {{𝑧 ∈ 1o𝜑}} → (𝑥𝑦) = ({{𝑧 ∈ 1o𝜑}} ∪ 𝑦))
1514eleq1d 2209 . . . . . . . 8 (𝑥 = {{𝑧 ∈ 1o𝜑}} → ((𝑥𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1o𝜑}} ∪ 𝑦) ∈ Fin))
16 uneq2 3228 . . . . . . . . 9 (𝑦 = {1o} → ({{𝑧 ∈ 1o𝜑}} ∪ 𝑦) = ({{𝑧 ∈ 1o𝜑}} ∪ {1o}))
1716eleq1d 2209 . . . . . . . 8 (𝑦 = {1o} → (({{𝑧 ∈ 1o𝜑}} ∪ 𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1o𝜑}} ∪ {1o}) ∈ Fin))
1815, 17rspc2v 2805 . . . . . . 7 (({{𝑧 ∈ 1o𝜑}} ∈ Fin ∧ {1o} ∈ Fin) → (∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥𝑦) ∈ Fin → ({{𝑧 ∈ 1o𝜑}} ∪ {1o}) ∈ Fin))
1910, 13, 18mp2an 423 . . . . . 6 (∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥𝑦) ∈ Fin → ({{𝑧 ∈ 1o𝜑}} ∪ {1o}) ∈ Fin)
203, 19ax-mp 5 . . . . 5 ({{𝑧 ∈ 1o𝜑}} ∪ {1o}) ∈ Fin
211, 20eqeltri 2213 . . . 4 {{𝑧 ∈ 1o𝜑}, 1o} ∈ Fin
228elexi 2701 . . . . 5 {𝑧 ∈ 1o𝜑} ∈ V
2322prid1 3636 . . . 4 {𝑧 ∈ 1o𝜑} ∈ {{𝑧 ∈ 1o𝜑}, 1o}
2411elexi 2701 . . . . 5 1o ∈ V
2524prid2 3637 . . . 4 1o ∈ {{𝑧 ∈ 1o𝜑}, 1o}
26 fidceq 6770 . . . 4 (({{𝑧 ∈ 1o𝜑}, 1o} ∈ Fin ∧ {𝑧 ∈ 1o𝜑} ∈ {{𝑧 ∈ 1o𝜑}, 1o} ∧ 1o ∈ {{𝑧 ∈ 1o𝜑}, 1o}) → DECID {𝑧 ∈ 1o𝜑} = 1o)
2721, 23, 25, 26mp3an 1316 . . 3 DECID {𝑧 ∈ 1o𝜑} = 1o
28 exmiddc 822 . . 3 (DECID {𝑧 ∈ 1o𝜑} = 1o → ({𝑧 ∈ 1o𝜑} = 1o ∨ ¬ {𝑧 ∈ 1o𝜑} = 1o))
2927, 28ax-mp 5 . 2 ({𝑧 ∈ 1o𝜑} = 1o ∨ ¬ {𝑧 ∈ 1o𝜑} = 1o)
304eqeq2i 2151 . . . 4 ({𝑧 ∈ 1o𝜑} = 1o ↔ {𝑧 ∈ 1o𝜑} = {∅})
31 0ex 4062 . . . . 5 ∅ ∈ V
32 biidd 171 . . . . 5 (𝑧 = ∅ → (𝜑𝜑))
3331, 32rabsnt 3605 . . . 4 ({𝑧 ∈ 1o𝜑} = {∅} → 𝜑)
3430, 33sylbi 120 . . 3 ({𝑧 ∈ 1o𝜑} = 1o𝜑)
35 df-rab 2426 . . . . 5 {𝑧 ∈ 1o𝜑} = {𝑧 ∣ (𝑧 ∈ 1o𝜑)}
36 iba 298 . . . . . 6 (𝜑 → (𝑧 ∈ 1o ↔ (𝑧 ∈ 1o𝜑)))
3736abbi2dv 2259 . . . . 5 (𝜑 → 1o = {𝑧 ∣ (𝑧 ∈ 1o𝜑)})
3835, 37eqtr4id 2192 . . . 4 (𝜑 → {𝑧 ∈ 1o𝜑} = 1o)
3938con3i 622 . . 3 (¬ {𝑧 ∈ 1o𝜑} = 1o → ¬ 𝜑)
4034, 39orim12i 749 . 2 (({𝑧 ∈ 1o𝜑} = 1o ∨ ¬ {𝑧 ∈ 1o𝜑} = 1o) → (𝜑 ∨ ¬ 𝜑))
4129, 40ax-mp 5 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 698  DECID wdc 820   = wceq 1332  wcel 1481  {cab 2126  wral 2417  {crab 2421  cun 3073  c0 3367  {csn 3531  {cpr 3532  Oncon0 4292  ωcom 4511  1oc1o 6313  Fincfn 6641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-br 3937  df-opab 3997  df-tr 4034  df-id 4222  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-1o 6320  df-en 6642  df-fin 6644
This theorem is referenced by: (None)
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