Theorem 0ct 7166

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 6493	. . . . 5 ⊢ ∅ ∈ 1o
2		djurcl 7111	. . . . 5 ⊢ (∅ ∈ 1o → (inr‘∅) ∈ (∅ ⊔ 1o))
3	1, 2	ax-mp 5	. . . 4 ⊢ (inr‘∅) ∈ (∅ ⊔ 1o)
4	3	fconst6 5453	. . 3 ⊢ (ω × {(inr‘∅)}):ω⟶(∅ ⊔ 1o)
5		peano1 4626	. . . . 5 ⊢ ∅ ∈ ω
6		rex0 3464	. . . . . . . . 9 ⊢ ¬ ∃𝑤 ∈ ∅ 𝑦 = (inl‘𝑤)
7		djur 7128	. . . . . . . . . . 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 6482	. . . . . . . . 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 4156	. . . . . . . 8 ⊢ ∅ ∈ V
15		fveq2 5554	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (inr‘𝑤) = (inr‘∅))
16	15	eqeq2d 2205	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑦 = (inr‘𝑤) ↔ 𝑦 = (inr‘∅)))
17	14, 16	rexsn 3662	. . . . . . 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 5774	. . . . . . 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 5554	. . . . . 6 ⊢ (𝑧 = ∅ → ((ω × {(inr‘∅)})‘𝑧) = ((ω × {(inr‘∅)})‘∅))
24	23	rspceeqv 2882	. . . . 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 5705	. . 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 4625	. . . 4 ⊢ ω ∈ V
30	19	snex 4214	. . . 4 ⊢ {(inr‘∅)} ∈ V
31	29, 30	xpex 4774	. . 3 ⊢ (ω × {(inr‘∅)}) ∈ V
32		foeq1 5472	. . 3 ⊢ (𝑓 = (ω × {(inr‘∅)}) → (𝑓:ω–onto→(∅ ⊔ 1o) ↔ (ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o)))
33	31, 32	spcev 2855	. 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 3446  {csn 3618  ωcom 4622   × cxp 4657  ⟶wf 5250  –onto→wfo 5252  ‘cfv 5254  1_oc1o 6462   ⊔ cdju 7096  inlcinl 7104  inrcinr 7105

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 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-iinf 4620

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 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-1st 6193  df-2nd 6194  df-1o 6469  df-dju 7097  df-inl 7106  df-inr 7107

This theorem is referenced by:  enumct  7174