Theorem 0ct 7168

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.

Step	Hyp	Ref	Expression
1		0lt1o 6495	. . . . 5 ⊢ ∅ ∈ 1o
2		djurcl 7113	. . . . 5 ⊢ (∅ ∈ 1o → (inr‘∅) ∈ (∅ ⊔ 1o))
3	1, 2	ax-mp 5	. . . 4 ⊢ (inr‘∅) ∈ (∅ ⊔ 1o)
4	3	fconst6 5454	. . 3 ⊢ (ω × {(inr‘∅)}):ω⟶(∅ ⊔ 1o)
5		peano1 4627	. . . . 5 ⊢ ∅ ∈ ω
6		rex0 3465	. . . . . . . . 9 ⊢ ¬ ∃𝑤 ∈ ∅ 𝑦 = (inl‘𝑤)
7		djur 7130	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (∅ ⊔ 1o) ↔ (∃𝑤 ∈ ∅ 𝑦 = (inl‘𝑤) ∨ ∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤)))
8	7	biimpi 120	. . . . . . . . . 10 ⊢ (𝑦 ∈ (∅ ⊔ 1o) → (∃𝑤 ∈ ∅ 𝑦 = (inl‘𝑤) ∨ ∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤)))
9	8	ord 725	. . . . . . . . 9 ⊢ (𝑦 ∈ (∅ ⊔ 1o) → (¬ ∃𝑤 ∈ ∅ 𝑦 = (inl‘𝑤) → ∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤)))
10	6, 9	mpi 15	. . . . . . . 8 ⊢ (𝑦 ∈ (∅ ⊔ 1o) → ∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤))
11		df1o2 6484	. . . . . . . . 9 ⊢ 1o = {∅}
12	11	rexeqi 2695	. . . . . . . 8 ⊢ (∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤) ↔ ∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤))
13	10, 12	sylib 122	. . . . . . 7 ⊢ (𝑦 ∈ (∅ ⊔ 1o) → ∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤))
14		0ex 4157	. . . . . . . 8 ⊢ ∅ ∈ V
15		fveq2 5555	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (inr‘𝑤) = (inr‘∅))
16	15	eqeq2d 2205	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑦 = (inr‘𝑤) ↔ 𝑦 = (inr‘∅)))
17	14, 16	rexsn 3663	. . . . . . 7 ⊢ (∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤) ↔ 𝑦 = (inr‘∅))
18	13, 17	sylib 122	. . . . . 6 ⊢ (𝑦 ∈ (∅ ⊔ 1o) → 𝑦 = (inr‘∅))
19	3	elexi 2772	. . . . . . . 8 ⊢ (inr‘∅) ∈ V
20	19	fvconst2 5775	. . . . . . 7 ⊢ (∅ ∈ ω → ((ω × {(inr‘∅)})‘∅) = (inr‘∅))
21	5, 20	ax-mp 5	. . . . . 6 ⊢ ((ω × {(inr‘∅)})‘∅) = (inr‘∅)
22	18, 21	eqtr4di 2244	. . . . 5 ⊢ (𝑦 ∈ (∅ ⊔ 1o) → 𝑦 = ((ω × {(inr‘∅)})‘∅))
23		fveq2 5555	. . . . . 6 ⊢ (𝑧 = ∅ → ((ω × {(inr‘∅)})‘𝑧) = ((ω × {(inr‘∅)})‘∅))
24	23	rspceeqv 2883	. . . . 5 ⊢ ((∅ ∈ ω ∧ 𝑦 = ((ω × {(inr‘∅)})‘∅)) → ∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧))
25	5, 22, 24	sylancr 414	. . . 4 ⊢ (𝑦 ∈ (∅ ⊔ 1o) → ∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧))
26	25	rgen 2547	. . 3 ⊢ ∀𝑦 ∈ (∅ ⊔ 1o)∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧)
27		dffo3 5706	. . 3 ⊢ ((ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o) ↔ ((ω × {(inr‘∅)}):ω⟶(∅ ⊔ 1o) ∧ ∀𝑦 ∈ (∅ ⊔ 1o)∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧)))
28	4, 26, 27	mpbir2an 944	. 2 ⊢ (ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o)
29		omex 4626	. . . 4 ⊢ ω ∈ V
30	19	snex 4215	. . . 4 ⊢ {(inr‘∅)} ∈ V
31	29, 30	xpex 4775	. . 3 ⊢ (ω × {(inr‘∅)}) ∈ V
32		foeq1 5473	. . 3 ⊢ (𝑓 = (ω × {(inr‘∅)}) → (𝑓:ω–onto→(∅ ⊔ 1o) ↔ (ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o)))
33	31, 32	spcev 2856	. 2 ⊢ ((ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o) → ∃𝑓 𝑓:ω–onto→(∅ ⊔ 1o))
34	28, 33	ax-mp 5	1 ⊢ ∃𝑓 𝑓:ω–onto→(∅ ⊔ 1o)

Syntax hints:  ¬ wn 3   ∨ wo 709   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  ∅c0 3447  {csn 3619  ωcom 4623   × cxp 4658  ⟶wf 5251  –onto→wfo 5253  ‘cfv 5255  1_oc1o 6464   ⊔ cdju 7098  inlcinl 7106  inrcinr 7107

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-iinf 4621

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6195  df-2nd 6196  df-1o 6471  df-dju 7099  df-inl 7108  df-inr 7109

This theorem is referenced by:  enumct  7176