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Theorem 0ct 6996
 Description: The empty set is countable. Remark of [BauerSwan], p. 14:3 which also has the definition of countable used here. (Contributed by Jim Kingdon, 13-Mar-2023.)
Assertion
Ref Expression
0ct 𝑓 𝑓:ω–onto→(∅ ⊔ 1o)

Proof of Theorem 0ct
Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0lt1o 6341 . . . . 5 ∅ ∈ 1o
2 djurcl 6941 . . . . 5 (∅ ∈ 1o → (inr‘∅) ∈ (∅ ⊔ 1o))
31, 2ax-mp 5 . . . 4 (inr‘∅) ∈ (∅ ⊔ 1o)
43fconst6 5326 . . 3 (ω × {(inr‘∅)}):ω⟶(∅ ⊔ 1o)
5 peano1 4512 . . . . 5 ∅ ∈ ω
6 rex0 3381 . . . . . . . . 9 ¬ ∃𝑤 ∈ ∅ 𝑦 = (inl‘𝑤)
7 djur 6958 . . . . . . . . . . 11 (𝑦 ∈ (∅ ⊔ 1o) ↔ (∃𝑤 ∈ ∅ 𝑦 = (inl‘𝑤) ∨ ∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤)))
87biimpi 119 . . . . . . . . . 10 (𝑦 ∈ (∅ ⊔ 1o) → (∃𝑤 ∈ ∅ 𝑦 = (inl‘𝑤) ∨ ∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤)))
98ord 714 . . . . . . . . 9 (𝑦 ∈ (∅ ⊔ 1o) → (¬ ∃𝑤 ∈ ∅ 𝑦 = (inl‘𝑤) → ∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤)))
106, 9mpi 15 . . . . . . . 8 (𝑦 ∈ (∅ ⊔ 1o) → ∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤))
11 df1o2 6330 . . . . . . . . 9 1o = {∅}
1211rexeqi 2632 . . . . . . . 8 (∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤) ↔ ∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤))
1310, 12sylib 121 . . . . . . 7 (𝑦 ∈ (∅ ⊔ 1o) → ∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤))
14 0ex 4059 . . . . . . . 8 ∅ ∈ V
15 fveq2 5425 . . . . . . . . 9 (𝑤 = ∅ → (inr‘𝑤) = (inr‘∅))
1615eqeq2d 2152 . . . . . . . 8 (𝑤 = ∅ → (𝑦 = (inr‘𝑤) ↔ 𝑦 = (inr‘∅)))
1714, 16rexsn 3571 . . . . . . 7 (∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤) ↔ 𝑦 = (inr‘∅))
1813, 17sylib 121 . . . . . 6 (𝑦 ∈ (∅ ⊔ 1o) → 𝑦 = (inr‘∅))
193elexi 2699 . . . . . . . 8 (inr‘∅) ∈ V
2019fvconst2 5640 . . . . . . 7 (∅ ∈ ω → ((ω × {(inr‘∅)})‘∅) = (inr‘∅))
215, 20ax-mp 5 . . . . . 6 ((ω × {(inr‘∅)})‘∅) = (inr‘∅)
2218, 21eqtr4di 2191 . . . . 5 (𝑦 ∈ (∅ ⊔ 1o) → 𝑦 = ((ω × {(inr‘∅)})‘∅))
23 fveq2 5425 . . . . . 6 (𝑧 = ∅ → ((ω × {(inr‘∅)})‘𝑧) = ((ω × {(inr‘∅)})‘∅))
2423rspceeqv 2808 . . . . 5 ((∅ ∈ ω ∧ 𝑦 = ((ω × {(inr‘∅)})‘∅)) → ∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧))
255, 22, 24sylancr 411 . . . 4 (𝑦 ∈ (∅ ⊔ 1o) → ∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧))
2625rgen 2486 . . 3 𝑦 ∈ (∅ ⊔ 1o)∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧)
27 dffo3 5571 . . 3 ((ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o) ↔ ((ω × {(inr‘∅)}):ω⟶(∅ ⊔ 1o) ∧ ∀𝑦 ∈ (∅ ⊔ 1o)∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧)))
284, 26, 27mpbir2an 927 . 2 (ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o)
29 omex 4511 . . . 4 ω ∈ V
3019snex 4113 . . . 4 {(inr‘∅)} ∈ V
3129, 30xpex 4658 . . 3 (ω × {(inr‘∅)}) ∈ V
32 foeq1 5345 . . 3 (𝑓 = (ω × {(inr‘∅)}) → (𝑓:ω–onto→(∅ ⊔ 1o) ↔ (ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o)))
3331, 32spcev 2781 . 2 ((ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o) → ∃𝑓 𝑓:ω–onto→(∅ ⊔ 1o))
3428, 33ax-mp 5 1 𝑓 𝑓:ω–onto→(∅ ⊔ 1o)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∨ wo 698   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  ∅c0 3364  {csn 3528  ωcom 4508   × cxp 4541  ⟶wf 5123  –onto→wfo 5125  ‘cfv 5127  1oc1o 6310   ⊔ cdju 6926  inlcinl 6934  inrcinr 6935 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 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-iinf 4506 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  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-ral 2422  df-rex 2423  df-v 2689  df-sbc 2911  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-id 4219  df-iord 4292  df-on 4294  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-1st 6042  df-2nd 6043  df-1o 6317  df-dju 6927  df-inl 6936  df-inr 6937 This theorem is referenced by:  enumct  7004
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