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Theorem 0ct 7135
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 6464 . . . . 5 ∅ ∈ 1o
2 djurcl 7080 . . . . 5 (∅ ∈ 1o → (inr‘∅) ∈ (∅ ⊔ 1o))
31, 2ax-mp 5 . . . 4 (inr‘∅) ∈ (∅ ⊔ 1o)
43fconst6 5434 . . 3 (ω × {(inr‘∅)}):ω⟶(∅ ⊔ 1o)
5 peano1 4611 . . . . 5 ∅ ∈ ω
6 rex0 3455 . . . . . . . . 9 ¬ ∃𝑤 ∈ ∅ 𝑦 = (inl‘𝑤)
7 djur 7097 . . . . . . . . . . 11 (𝑦 ∈ (∅ ⊔ 1o) ↔ (∃𝑤 ∈ ∅ 𝑦 = (inl‘𝑤) ∨ ∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤)))
87biimpi 120 . . . . . . . . . 10 (𝑦 ∈ (∅ ⊔ 1o) → (∃𝑤 ∈ ∅ 𝑦 = (inl‘𝑤) ∨ ∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤)))
98ord 725 . . . . . . . . 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 6453 . . . . . . . . 9 1o = {∅}
1211rexeqi 2691 . . . . . . . 8 (∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤) ↔ ∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤))
1310, 12sylib 122 . . . . . . 7 (𝑦 ∈ (∅ ⊔ 1o) → ∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤))
14 0ex 4145 . . . . . . . 8 ∅ ∈ V
15 fveq2 5534 . . . . . . . . 9 (𝑤 = ∅ → (inr‘𝑤) = (inr‘∅))
1615eqeq2d 2201 . . . . . . . 8 (𝑤 = ∅ → (𝑦 = (inr‘𝑤) ↔ 𝑦 = (inr‘∅)))
1714, 16rexsn 3651 . . . . . . 7 (∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤) ↔ 𝑦 = (inr‘∅))
1813, 17sylib 122 . . . . . 6 (𝑦 ∈ (∅ ⊔ 1o) → 𝑦 = (inr‘∅))
193elexi 2764 . . . . . . . 8 (inr‘∅) ∈ V
2019fvconst2 5752 . . . . . . 7 (∅ ∈ ω → ((ω × {(inr‘∅)})‘∅) = (inr‘∅))
215, 20ax-mp 5 . . . . . 6 ((ω × {(inr‘∅)})‘∅) = (inr‘∅)
2218, 21eqtr4di 2240 . . . . 5 (𝑦 ∈ (∅ ⊔ 1o) → 𝑦 = ((ω × {(inr‘∅)})‘∅))
23 fveq2 5534 . . . . . 6 (𝑧 = ∅ → ((ω × {(inr‘∅)})‘𝑧) = ((ω × {(inr‘∅)})‘∅))
2423rspceeqv 2874 . . . . 5 ((∅ ∈ ω ∧ 𝑦 = ((ω × {(inr‘∅)})‘∅)) → ∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧))
255, 22, 24sylancr 414 . . . 4 (𝑦 ∈ (∅ ⊔ 1o) → ∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧))
2625rgen 2543 . . 3 𝑦 ∈ (∅ ⊔ 1o)∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧)
27 dffo3 5683 . . 3 ((ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o) ↔ ((ω × {(inr‘∅)}):ω⟶(∅ ⊔ 1o) ∧ ∀𝑦 ∈ (∅ ⊔ 1o)∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧)))
284, 26, 27mpbir2an 944 . 2 (ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o)
29 omex 4610 . . . 4 ω ∈ V
3019snex 4203 . . . 4 {(inr‘∅)} ∈ V
3129, 30xpex 4759 . . 3 (ω × {(inr‘∅)}) ∈ V
32 foeq1 5453 . . 3 (𝑓 = (ω × {(inr‘∅)}) → (𝑓:ω–onto→(∅ ⊔ 1o) ↔ (ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o)))
3331, 32spcev 2847 . 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 709   = wceq 1364  wex 1503  wcel 2160  wral 2468  wrex 2469  c0 3437  {csn 3607  ωcom 4607   × cxp 4642  wf 5231  ontowfo 5233  cfv 5235  1oc1o 6433  cdju 7065  inlcinl 7073  inrcinr 7074
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6164  df-2nd 6165  df-1o 6440  df-dju 7066  df-inl 7075  df-inr 7076
This theorem is referenced by:  enumct  7143
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