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

Theorem unfiexmid 6735
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 3481 . . . . 5 {{𝑧 ∈ 1o𝜑}, 1o} = ({{𝑧 ∈ 1o𝜑}} ∪ {1o})
2 unfiexmid.1 . . . . . . 7 ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥𝑦) ∈ Fin)
32rgen2a 2445 . . . . . 6 𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥𝑦) ∈ Fin
4 df1o2 6256 . . . . . . . . . 10 1o = {∅}
5 rabeq 2633 . . . . . . . . . 10 (1o = {∅} → {𝑧 ∈ 1o𝜑} = {𝑧 ∈ {∅} ∣ 𝜑})
64, 5ax-mp 7 . . . . . . . . 9 {𝑧 ∈ 1o𝜑} = {𝑧 ∈ {∅} ∣ 𝜑}
7 ordtriexmidlem 4373 . . . . . . . . 9 {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
86, 7eqeltri 2172 . . . . . . . 8 {𝑧 ∈ 1o𝜑} ∈ On
9 snfig 6638 . . . . . . . 8 ({𝑧 ∈ 1o𝜑} ∈ On → {{𝑧 ∈ 1o𝜑}} ∈ Fin)
108, 9ax-mp 7 . . . . . . 7 {{𝑧 ∈ 1o𝜑}} ∈ Fin
11 1onn 6346 . . . . . . . 8 1o ∈ ω
12 snfig 6638 . . . . . . . 8 (1o ∈ ω → {1o} ∈ Fin)
1311, 12ax-mp 7 . . . . . . 7 {1o} ∈ Fin
14 uneq1 3170 . . . . . . . . 9 (𝑥 = {{𝑧 ∈ 1o𝜑}} → (𝑥𝑦) = ({{𝑧 ∈ 1o𝜑}} ∪ 𝑦))
1514eleq1d 2168 . . . . . . . 8 (𝑥 = {{𝑧 ∈ 1o𝜑}} → ((𝑥𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1o𝜑}} ∪ 𝑦) ∈ Fin))
16 uneq2 3171 . . . . . . . . 9 (𝑦 = {1o} → ({{𝑧 ∈ 1o𝜑}} ∪ 𝑦) = ({{𝑧 ∈ 1o𝜑}} ∪ {1o}))
1716eleq1d 2168 . . . . . . . 8 (𝑦 = {1o} → (({{𝑧 ∈ 1o𝜑}} ∪ 𝑦) ∈ Fin ↔ ({{𝑧 ∈ 1o𝜑}} ∪ {1o}) ∈ Fin))
1815, 17rspc2v 2756 . . . . . . 7 (({{𝑧 ∈ 1o𝜑}} ∈ Fin ∧ {1o} ∈ Fin) → (∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥𝑦) ∈ Fin → ({{𝑧 ∈ 1o𝜑}} ∪ {1o}) ∈ Fin))
1910, 13, 18mp2an 420 . . . . . 6 (∀𝑥 ∈ Fin ∀𝑦 ∈ Fin (𝑥𝑦) ∈ Fin → ({{𝑧 ∈ 1o𝜑}} ∪ {1o}) ∈ Fin)
203, 19ax-mp 7 . . . . 5 ({{𝑧 ∈ 1o𝜑}} ∪ {1o}) ∈ Fin
211, 20eqeltri 2172 . . . 4 {{𝑧 ∈ 1o𝜑}, 1o} ∈ Fin
228elexi 2653 . . . . 5 {𝑧 ∈ 1o𝜑} ∈ V
2322prid1 3576 . . . 4 {𝑧 ∈ 1o𝜑} ∈ {{𝑧 ∈ 1o𝜑}, 1o}
2411elexi 2653 . . . . 5 1o ∈ V
2524prid2 3577 . . . 4 1o ∈ {{𝑧 ∈ 1o𝜑}, 1o}
26 fidceq 6692 . . . 4 (({{𝑧 ∈ 1o𝜑}, 1o} ∈ Fin ∧ {𝑧 ∈ 1o𝜑} ∈ {{𝑧 ∈ 1o𝜑}, 1o} ∧ 1o ∈ {{𝑧 ∈ 1o𝜑}, 1o}) → DECID {𝑧 ∈ 1o𝜑} = 1o)
2721, 23, 25, 26mp3an 1283 . . 3 DECID {𝑧 ∈ 1o𝜑} = 1o
28 exmiddc 788 . . 3 (DECID {𝑧 ∈ 1o𝜑} = 1o → ({𝑧 ∈ 1o𝜑} = 1o ∨ ¬ {𝑧 ∈ 1o𝜑} = 1o))
2927, 28ax-mp 7 . 2 ({𝑧 ∈ 1o𝜑} = 1o ∨ ¬ {𝑧 ∈ 1o𝜑} = 1o)
304eqeq2i 2110 . . . 4 ({𝑧 ∈ 1o𝜑} = 1o ↔ {𝑧 ∈ 1o𝜑} = {∅})
31 0ex 3995 . . . . 5 ∅ ∈ V
32 biidd 171 . . . . 5 (𝑧 = ∅ → (𝜑𝜑))
3331, 32rabsnt 3545 . . . 4 ({𝑧 ∈ 1o𝜑} = {∅} → 𝜑)
3430, 33sylbi 120 . . 3 ({𝑧 ∈ 1o𝜑} = 1o𝜑)
35 iba 296 . . . . . 6 (𝜑 → (𝑧 ∈ 1o ↔ (𝑧 ∈ 1o𝜑)))
3635abbi2dv 2218 . . . . 5 (𝜑 → 1o = {𝑧 ∣ (𝑧 ∈ 1o𝜑)})
37 df-rab 2384 . . . . 5 {𝑧 ∈ 1o𝜑} = {𝑧 ∣ (𝑧 ∈ 1o𝜑)}
3836, 37syl6reqr 2151 . . . 4 (𝜑 → {𝑧 ∈ 1o𝜑} = 1o)
3938con3i 602 . . 3 (¬ {𝑧 ∈ 1o𝜑} = 1o → ¬ 𝜑)
4034, 39orim12i 717 . 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 103  wo 670  DECID wdc 786   = wceq 1299  wcel 1448  {cab 2086  wral 2375  {crab 2379  cun 3019  c0 3310  {csn 3474  {cpr 3475  Oncon0 4223  ωcom 4442  1oc1o 6236  Fincfn 6564
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-br 3876  df-opab 3930  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-1o 6243  df-en 6565  df-fin 6567
This theorem is referenced by: (None)
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