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

Theorem unfiexmid 6917
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 3600 . . . . 5 {{𝑧 ∈ 1o𝜑}, 1o} = ({{𝑧 ∈ 1o𝜑}} ∪ {1o})
2 unfiexmid.1 . . . . . . 7 ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥𝑦) ∈ Fin)
32rgen2a 2531 . . . . . 6 𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥𝑦) ∈ Fin
4 df1o2 6430 . . . . . . . . . 10 1o = {∅}
5 rabeq 2730 . . . . . . . . . 10 (1o = {∅} → {𝑧 ∈ 1o𝜑} = {𝑧 ∈ {∅} ∣ 𝜑})
64, 5ax-mp 5 . . . . . . . . 9 {𝑧 ∈ 1o𝜑} = {𝑧 ∈ {∅} ∣ 𝜑}
7 ordtriexmidlem 4519 . . . . . . . . 9 {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
86, 7eqeltri 2250 . . . . . . . 8 {𝑧 ∈ 1o𝜑} ∈ On
9 snfig 6814 . . . . . . . 8 ({𝑧 ∈ 1o𝜑} ∈ On → {{𝑧 ∈ 1o𝜑}} ∈ Fin)
108, 9ax-mp 5 . . . . . . 7 {{𝑧 ∈ 1o𝜑}} ∈ Fin
11 1onn 6521 . . . . . . . 8 1o ∈ ω
12 snfig 6814 . . . . . . . 8 (1o ∈ ω → {1o} ∈ Fin)
1311, 12ax-mp 5 . . . . . . 7 {1o} ∈ Fin
14 uneq1 3283 . . . . . . . . 9 (𝑥 = {{𝑧 ∈ 1o𝜑}} → (𝑥𝑦) = ({{𝑧 ∈ 1o𝜑}} ∪ 𝑦))
1514eleq1d 2246 . . . . . . . 8 (𝑥 = {{𝑧 ∈ 1o𝜑}} → ((𝑥𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1o𝜑}} ∪ 𝑦) ∈ Fin))
16 uneq2 3284 . . . . . . . . 9 (𝑦 = {1o} → ({{𝑧 ∈ 1o𝜑}} ∪ 𝑦) = ({{𝑧 ∈ 1o𝜑}} ∪ {1o}))
1716eleq1d 2246 . . . . . . . 8 (𝑦 = {1o} → (({{𝑧 ∈ 1o𝜑}} ∪ 𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1o𝜑}} ∪ {1o}) ∈ Fin))
1815, 17rspc2v 2855 . . . . . . 7 (({{𝑧 ∈ 1o𝜑}} ∈ Fin ∧ {1o} ∈ Fin) → (∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥𝑦) ∈ Fin → ({{𝑧 ∈ 1o𝜑}} ∪ {1o}) ∈ Fin))
1910, 13, 18mp2an 426 . . . . . 6 (∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥𝑦) ∈ Fin → ({{𝑧 ∈ 1o𝜑}} ∪ {1o}) ∈ Fin)
203, 19ax-mp 5 . . . . 5 ({{𝑧 ∈ 1o𝜑}} ∪ {1o}) ∈ Fin
211, 20eqeltri 2250 . . . 4 {{𝑧 ∈ 1o𝜑}, 1o} ∈ Fin
228elexi 2750 . . . . 5 {𝑧 ∈ 1o𝜑} ∈ V
2322prid1 3699 . . . 4 {𝑧 ∈ 1o𝜑} ∈ {{𝑧 ∈ 1o𝜑}, 1o}
2411elexi 2750 . . . . 5 1o ∈ V
2524prid2 3700 . . . 4 1o ∈ {{𝑧 ∈ 1o𝜑}, 1o}
26 fidceq 6869 . . . 4 (({{𝑧 ∈ 1o𝜑}, 1o} ∈ Fin ∧ {𝑧 ∈ 1o𝜑} ∈ {{𝑧 ∈ 1o𝜑}, 1o} ∧ 1o ∈ {{𝑧 ∈ 1o𝜑}, 1o}) → DECID {𝑧 ∈ 1o𝜑} = 1o)
2721, 23, 25, 26mp3an 1337 . . 3 DECID {𝑧 ∈ 1o𝜑} = 1o
28 exmiddc 836 . . 3 (DECID {𝑧 ∈ 1o𝜑} = 1o → ({𝑧 ∈ 1o𝜑} = 1o ∨ ¬ {𝑧 ∈ 1o𝜑} = 1o))
2927, 28ax-mp 5 . 2 ({𝑧 ∈ 1o𝜑} = 1o ∨ ¬ {𝑧 ∈ 1o𝜑} = 1o)
304eqeq2i 2188 . . . 4 ({𝑧 ∈ 1o𝜑} = 1o ↔ {𝑧 ∈ 1o𝜑} = {∅})
31 0ex 4131 . . . . 5 ∅ ∈ V
32 biidd 172 . . . . 5 (𝑧 = ∅ → (𝜑𝜑))
3331, 32rabsnt 3668 . . . 4 ({𝑧 ∈ 1o𝜑} = {∅} → 𝜑)
3430, 33sylbi 121 . . 3 ({𝑧 ∈ 1o𝜑} = 1o𝜑)
35 df-rab 2464 . . . . 5 {𝑧 ∈ 1o𝜑} = {𝑧 ∣ (𝑧 ∈ 1o𝜑)}
36 iba 300 . . . . . 6 (𝜑 → (𝑧 ∈ 1o ↔ (𝑧 ∈ 1o𝜑)))
3736abbi2dv 2296 . . . . 5 (𝜑 → 1o = {𝑧 ∣ (𝑧 ∈ 1o𝜑)})
3835, 37eqtr4id 2229 . . . 4 (𝜑 → {𝑧 ∈ 1o𝜑} = 1o)
3938con3i 632 . . 3 (¬ {𝑧 ∈ 1o𝜑} = 1o → ¬ 𝜑)
4034, 39orim12i 759 . 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 104  wo 708  DECID wdc 834   = wceq 1353  wcel 2148  {cab 2163  wral 2455  {crab 2459  cun 3128  c0 3423  {csn 3593  {cpr 3594  Oncon0 4364  ωcom 4590  1oc1o 6410  Fincfn 6740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-1o 6417  df-en 6741  df-fin 6743
This theorem is referenced by: (None)
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