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

Theorem unfiexmid 6462
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 3423 . . . . 5 {{𝑧 ∈ 1𝑜𝜑}, 1𝑜} = ({{𝑧 ∈ 1𝑜𝜑}} ∪ {1𝑜})
2 unfiexmid.1 . . . . . . 7 ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥𝑦) ∈ Fin)
32rgen2a 2422 . . . . . 6 𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥𝑦) ∈ Fin
4 df1o2 6097 . . . . . . . . . 10 1𝑜 = {∅}
5 rabeq 2602 . . . . . . . . . 10 (1𝑜 = {∅} → {𝑧 ∈ 1𝑜𝜑} = {𝑧 ∈ {∅} ∣ 𝜑})
64, 5ax-mp 7 . . . . . . . . 9 {𝑧 ∈ 1𝑜𝜑} = {𝑧 ∈ {∅} ∣ 𝜑}
7 ordtriexmidlem 4291 . . . . . . . . 9 {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
86, 7eqeltri 2155 . . . . . . . 8 {𝑧 ∈ 1𝑜𝜑} ∈ On
9 snfig 6380 . . . . . . . 8 ({𝑧 ∈ 1𝑜𝜑} ∈ On → {{𝑧 ∈ 1𝑜𝜑}} ∈ Fin)
108, 9ax-mp 7 . . . . . . 7 {{𝑧 ∈ 1𝑜𝜑}} ∈ Fin
11 1onn 6180 . . . . . . . 8 1𝑜 ∈ ω
12 snfig 6380 . . . . . . . 8 (1𝑜 ∈ ω → {1𝑜} ∈ Fin)
1311, 12ax-mp 7 . . . . . . 7 {1𝑜} ∈ Fin
14 uneq1 3129 . . . . . . . . 9 (𝑥 = {{𝑧 ∈ 1𝑜𝜑}} → (𝑥𝑦) = ({{𝑧 ∈ 1𝑜𝜑}} ∪ 𝑦))
1514eleq1d 2151 . . . . . . . 8 (𝑥 = {{𝑧 ∈ 1𝑜𝜑}} → ((𝑥𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1𝑜𝜑}} ∪ 𝑦) ∈ Fin))
16 uneq2 3130 . . . . . . . . 9 (𝑦 = {1𝑜} → ({{𝑧 ∈ 1𝑜𝜑}} ∪ 𝑦) = ({{𝑧 ∈ 1𝑜𝜑}} ∪ {1𝑜}))
1716eleq1d 2151 . . . . . . . 8 (𝑦 = {1𝑜} → (({{𝑧 ∈ 1𝑜𝜑}} ∪ 𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1𝑜𝜑}} ∪ {1𝑜}) ∈ Fin))
1815, 17rspc2v 2721 . . . . . . 7 (({{𝑧 ∈ 1𝑜𝜑}} ∈ Fin ∧ {1𝑜} ∈ Fin) → (∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥𝑦) ∈ Fin → ({{𝑧 ∈ 1𝑜𝜑}} ∪ {1𝑜}) ∈ Fin))
1910, 13, 18mp2an 417 . . . . . 6 (∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥𝑦) ∈ Fin → ({{𝑧 ∈ 1𝑜𝜑}} ∪ {1𝑜}) ∈ Fin)
203, 19ax-mp 7 . . . . 5 ({{𝑧 ∈ 1𝑜𝜑}} ∪ {1𝑜}) ∈ Fin
211, 20eqeltri 2155 . . . 4 {{𝑧 ∈ 1𝑜𝜑}, 1𝑜} ∈ Fin
228elexi 2620 . . . . 5 {𝑧 ∈ 1𝑜𝜑} ∈ V
2322prid1 3516 . . . 4 {𝑧 ∈ 1𝑜𝜑} ∈ {{𝑧 ∈ 1𝑜𝜑}, 1𝑜}
2411elexi 2620 . . . . 5 1𝑜 ∈ V
2524prid2 3517 . . . 4 1𝑜 ∈ {{𝑧 ∈ 1𝑜𝜑}, 1𝑜}
26 fidceq 6425 . . . 4 (({{𝑧 ∈ 1𝑜𝜑}, 1𝑜} ∈ Fin ∧ {𝑧 ∈ 1𝑜𝜑} ∈ {{𝑧 ∈ 1𝑜𝜑}, 1𝑜} ∧ 1𝑜 ∈ {{𝑧 ∈ 1𝑜𝜑}, 1𝑜}) → DECID {𝑧 ∈ 1𝑜𝜑} = 1𝑜)
2721, 23, 25, 26mp3an 1269 . . 3 DECID {𝑧 ∈ 1𝑜𝜑} = 1𝑜
28 exmiddc 778 . . 3 (DECID {𝑧 ∈ 1𝑜𝜑} = 1𝑜 → ({𝑧 ∈ 1𝑜𝜑} = 1𝑜 ∨ ¬ {𝑧 ∈ 1𝑜𝜑} = 1𝑜))
2927, 28ax-mp 7 . 2 ({𝑧 ∈ 1𝑜𝜑} = 1𝑜 ∨ ¬ {𝑧 ∈ 1𝑜𝜑} = 1𝑜)
304eqeq2i 2093 . . . 4 ({𝑧 ∈ 1𝑜𝜑} = 1𝑜 ↔ {𝑧 ∈ 1𝑜𝜑} = {∅})
31 0ex 3925 . . . . 5 ∅ ∈ V
32 biidd 170 . . . . 5 (𝑧 = ∅ → (𝜑𝜑))
3331, 32rabsnt 3485 . . . 4 ({𝑧 ∈ 1𝑜𝜑} = {∅} → 𝜑)
3430, 33sylbi 119 . . 3 ({𝑧 ∈ 1𝑜𝜑} = 1𝑜𝜑)
35 iba 294 . . . . . 6 (𝜑 → (𝑧 ∈ 1𝑜 ↔ (𝑧 ∈ 1𝑜𝜑)))
3635abbi2dv 2201 . . . . 5 (𝜑 → 1𝑜 = {𝑧 ∣ (𝑧 ∈ 1𝑜𝜑)})
37 df-rab 2362 . . . . 5 {𝑧 ∈ 1𝑜𝜑} = {𝑧 ∣ (𝑧 ∈ 1𝑜𝜑)}
3836, 37syl6reqr 2134 . . . 4 (𝜑 → {𝑧 ∈ 1𝑜𝜑} = 1𝑜)
3938con3i 595 . . 3 (¬ {𝑧 ∈ 1𝑜𝜑} = 1𝑜 → ¬ 𝜑)
4034, 39orim12i 709 . 2 (({𝑧 ∈ 1𝑜𝜑} = 1𝑜 ∨ ¬ {𝑧 ∈ 1𝑜𝜑} = 1𝑜) → (𝜑 ∨ ¬ 𝜑))
4129, 40ax-mp 7 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 662  DECID wdc 776   = wceq 1285  wcel 1434  {cab 2069  wral 2353  {crab 2357  cun 2980  c0 3267  {csn 3416  {cpr 3417  Oncon0 4146  ωcom 4359  1𝑜c1o 6078  Fincfn 6308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-br 3806  df-opab 3860  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-1o 6085  df-en 6309  df-fin 6311
This theorem is referenced by: (None)
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