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Theorem 0ct 7080
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 6416 . . . . 5 ∅ ∈ 1o
2 djurcl 7025 . . . . 5 (∅ ∈ 1o → (inr‘∅) ∈ (∅ ⊔ 1o))
31, 2ax-mp 5 . . . 4 (inr‘∅) ∈ (∅ ⊔ 1o)
43fconst6 5395 . . 3 (ω × {(inr‘∅)}):ω⟶(∅ ⊔ 1o)
5 peano1 4576 . . . . 5 ∅ ∈ ω
6 rex0 3431 . . . . . . . . 9 ¬ ∃𝑤 ∈ ∅ 𝑦 = (inl‘𝑤)
7 djur 7042 . . . . . . . . . . 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 719 . . . . . . . . 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 6405 . . . . . . . . 9 1o = {∅}
1211rexeqi 2670 . . . . . . . 8 (∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤) ↔ ∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤))
1310, 12sylib 121 . . . . . . 7 (𝑦 ∈ (∅ ⊔ 1o) → ∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤))
14 0ex 4114 . . . . . . . 8 ∅ ∈ V
15 fveq2 5494 . . . . . . . . 9 (𝑤 = ∅ → (inr‘𝑤) = (inr‘∅))
1615eqeq2d 2182 . . . . . . . 8 (𝑤 = ∅ → (𝑦 = (inr‘𝑤) ↔ 𝑦 = (inr‘∅)))
1714, 16rexsn 3625 . . . . . . 7 (∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤) ↔ 𝑦 = (inr‘∅))
1813, 17sylib 121 . . . . . 6 (𝑦 ∈ (∅ ⊔ 1o) → 𝑦 = (inr‘∅))
193elexi 2742 . . . . . . . 8 (inr‘∅) ∈ V
2019fvconst2 5709 . . . . . . 7 (∅ ∈ ω → ((ω × {(inr‘∅)})‘∅) = (inr‘∅))
215, 20ax-mp 5 . . . . . 6 ((ω × {(inr‘∅)})‘∅) = (inr‘∅)
2218, 21eqtr4di 2221 . . . . 5 (𝑦 ∈ (∅ ⊔ 1o) → 𝑦 = ((ω × {(inr‘∅)})‘∅))
23 fveq2 5494 . . . . . 6 (𝑧 = ∅ → ((ω × {(inr‘∅)})‘𝑧) = ((ω × {(inr‘∅)})‘∅))
2423rspceeqv 2852 . . . . 5 ((∅ ∈ ω ∧ 𝑦 = ((ω × {(inr‘∅)})‘∅)) → ∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧))
255, 22, 24sylancr 412 . . . 4 (𝑦 ∈ (∅ ⊔ 1o) → ∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧))
2625rgen 2523 . . 3 𝑦 ∈ (∅ ⊔ 1o)∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧)
27 dffo3 5640 . . 3 ((ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o) ↔ ((ω × {(inr‘∅)}):ω⟶(∅ ⊔ 1o) ∧ ∀𝑦 ∈ (∅ ⊔ 1o)∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧)))
284, 26, 27mpbir2an 937 . 2 (ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o)
29 omex 4575 . . . 4 ω ∈ V
3019snex 4169 . . . 4 {(inr‘∅)} ∈ V
3129, 30xpex 4724 . . 3 (ω × {(inr‘∅)}) ∈ V
32 foeq1 5414 . . 3 (𝑓 = (ω × {(inr‘∅)}) → (𝑓:ω–onto→(∅ ⊔ 1o) ↔ (ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o)))
3331, 32spcev 2825 . 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 703   = wceq 1348  wex 1485  wcel 2141  wral 2448  wrex 2449  c0 3414  {csn 3581  ωcom 4572   × cxp 4607  wf 5192  ontowfo 5194  cfv 5196  1oc1o 6385  cdju 7010  inlcinl 7018  inrcinr 7019
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-iinf 4570
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  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-ral 2453  df-rex 2454  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-mpt 4050  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-res 4621  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-1st 6116  df-2nd 6117  df-1o 6392  df-dju 7011  df-inl 7020  df-inr 7021
This theorem is referenced by:  enumct  7088
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