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

Theorem unfiexmid 6891
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 3588 . . . . 5 {{𝑧 ∈ 1o𝜑}, 1o} = ({{𝑧 ∈ 1o𝜑}} ∪ {1o})
2 unfiexmid.1 . . . . . . 7 ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥𝑦) ∈ Fin)
32rgen2a 2524 . . . . . 6 𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥𝑦) ∈ Fin
4 df1o2 6405 . . . . . . . . . 10 1o = {∅}
5 rabeq 2722 . . . . . . . . . 10 (1o = {∅} → {𝑧 ∈ 1o𝜑} = {𝑧 ∈ {∅} ∣ 𝜑})
64, 5ax-mp 5 . . . . . . . . 9 {𝑧 ∈ 1o𝜑} = {𝑧 ∈ {∅} ∣ 𝜑}
7 ordtriexmidlem 4501 . . . . . . . . 9 {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
86, 7eqeltri 2243 . . . . . . . 8 {𝑧 ∈ 1o𝜑} ∈ On
9 snfig 6788 . . . . . . . 8 ({𝑧 ∈ 1o𝜑} ∈ On → {{𝑧 ∈ 1o𝜑}} ∈ Fin)
108, 9ax-mp 5 . . . . . . 7 {{𝑧 ∈ 1o𝜑}} ∈ Fin
11 1onn 6496 . . . . . . . 8 1o ∈ ω
12 snfig 6788 . . . . . . . 8 (1o ∈ ω → {1o} ∈ Fin)
1311, 12ax-mp 5 . . . . . . 7 {1o} ∈ Fin
14 uneq1 3274 . . . . . . . . 9 (𝑥 = {{𝑧 ∈ 1o𝜑}} → (𝑥𝑦) = ({{𝑧 ∈ 1o𝜑}} ∪ 𝑦))
1514eleq1d 2239 . . . . . . . 8 (𝑥 = {{𝑧 ∈ 1o𝜑}} → ((𝑥𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1o𝜑}} ∪ 𝑦) ∈ Fin))
16 uneq2 3275 . . . . . . . . 9 (𝑦 = {1o} → ({{𝑧 ∈ 1o𝜑}} ∪ 𝑦) = ({{𝑧 ∈ 1o𝜑}} ∪ {1o}))
1716eleq1d 2239 . . . . . . . 8 (𝑦 = {1o} → (({{𝑧 ∈ 1o𝜑}} ∪ 𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1o𝜑}} ∪ {1o}) ∈ Fin))
1815, 17rspc2v 2847 . . . . . . 7 (({{𝑧 ∈ 1o𝜑}} ∈ Fin ∧ {1o} ∈ Fin) → (∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥𝑦) ∈ Fin → ({{𝑧 ∈ 1o𝜑}} ∪ {1o}) ∈ Fin))
1910, 13, 18mp2an 424 . . . . . 6 (∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥𝑦) ∈ Fin → ({{𝑧 ∈ 1o𝜑}} ∪ {1o}) ∈ Fin)
203, 19ax-mp 5 . . . . 5 ({{𝑧 ∈ 1o𝜑}} ∪ {1o}) ∈ Fin
211, 20eqeltri 2243 . . . 4 {{𝑧 ∈ 1o𝜑}, 1o} ∈ Fin
228elexi 2742 . . . . 5 {𝑧 ∈ 1o𝜑} ∈ V
2322prid1 3687 . . . 4 {𝑧 ∈ 1o𝜑} ∈ {{𝑧 ∈ 1o𝜑}, 1o}
2411elexi 2742 . . . . 5 1o ∈ V
2524prid2 3688 . . . 4 1o ∈ {{𝑧 ∈ 1o𝜑}, 1o}
26 fidceq 6843 . . . 4 (({{𝑧 ∈ 1o𝜑}, 1o} ∈ Fin ∧ {𝑧 ∈ 1o𝜑} ∈ {{𝑧 ∈ 1o𝜑}, 1o} ∧ 1o ∈ {{𝑧 ∈ 1o𝜑}, 1o}) → DECID {𝑧 ∈ 1o𝜑} = 1o)
2721, 23, 25, 26mp3an 1332 . . 3 DECID {𝑧 ∈ 1o𝜑} = 1o
28 exmiddc 831 . . 3 (DECID {𝑧 ∈ 1o𝜑} = 1o → ({𝑧 ∈ 1o𝜑} = 1o ∨ ¬ {𝑧 ∈ 1o𝜑} = 1o))
2927, 28ax-mp 5 . 2 ({𝑧 ∈ 1o𝜑} = 1o ∨ ¬ {𝑧 ∈ 1o𝜑} = 1o)
304eqeq2i 2181 . . . 4 ({𝑧 ∈ 1o𝜑} = 1o ↔ {𝑧 ∈ 1o𝜑} = {∅})
31 0ex 4114 . . . . 5 ∅ ∈ V
32 biidd 171 . . . . 5 (𝑧 = ∅ → (𝜑𝜑))
3331, 32rabsnt 3656 . . . 4 ({𝑧 ∈ 1o𝜑} = {∅} → 𝜑)
3430, 33sylbi 120 . . 3 ({𝑧 ∈ 1o𝜑} = 1o𝜑)
35 df-rab 2457 . . . . 5 {𝑧 ∈ 1o𝜑} = {𝑧 ∣ (𝑧 ∈ 1o𝜑)}
36 iba 298 . . . . . 6 (𝜑 → (𝑧 ∈ 1o ↔ (𝑧 ∈ 1o𝜑)))
3736abbi2dv 2289 . . . . 5 (𝜑 → 1o = {𝑧 ∣ (𝑧 ∈ 1o𝜑)})
3835, 37eqtr4id 2222 . . . 4 (𝜑 → {𝑧 ∈ 1o𝜑} = 1o)
3938con3i 627 . . 3 (¬ {𝑧 ∈ 1o𝜑} = 1o → ¬ 𝜑)
4034, 39orim12i 754 . 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 703  DECID wdc 829   = wceq 1348  wcel 2141  {cab 2156  wral 2448  {crab 2452  cun 3119  c0 3414  {csn 3581  {cpr 3582  Oncon0 4346  ωcom 4572  1oc1o 6385  Fincfn 6714
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-br 3988  df-opab 4049  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-1o 6392  df-en 6715  df-fin 6717
This theorem is referenced by: (None)
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