Theorem 0ct 7182

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 6507	. . . . 5 ⊢ ∅ ∈ 1o
2		djurcl 7127	. . . . 5 ⊢ (∅ ∈ 1o → (inr‘∅) ∈ (∅ ⊔ 1o))
3	1, 2	ax-mp 5	. . . 4 ⊢ (inr‘∅) ∈ (∅ ⊔ 1o)
4	3	fconst6 5460	. . 3 ⊢ (ω × {(inr‘∅)}):ω⟶(∅ ⊔ 1o)
5		peano1 4631	. . . . 5 ⊢ ∅ ∈ ω
6		rex0 3469	. . . . . . . . 9 ⊢ ¬ ∃𝑤 ∈ ∅ 𝑦 = (inl‘𝑤)
7		djur 7144	. . . . . . . . . . 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 6496	. . . . . . . . 9 ⊢ 1o = {∅}
12	11	rexeqi 2698	. . . . . . . 8 ⊢ (∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤) ↔ ∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤))
13	10, 12	sylib 122	. . . . . . 7 ⊢ (𝑦 ∈ (∅ ⊔ 1o) → ∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤))
14		0ex 4161	. . . . . . . 8 ⊢ ∅ ∈ V
15		fveq2 5561	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (inr‘𝑤) = (inr‘∅))
16	15	eqeq2d 2208	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑦 = (inr‘𝑤) ↔ 𝑦 = (inr‘∅)))
17	14, 16	rexsn 3667	. . . . . . 7 ⊢ (∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤) ↔ 𝑦 = (inr‘∅))
18	13, 17	sylib 122	. . . . . 6 ⊢ (𝑦 ∈ (∅ ⊔ 1o) → 𝑦 = (inr‘∅))
19	3	elexi 2775	. . . . . . . 8 ⊢ (inr‘∅) ∈ V
20	19	fvconst2 5781	. . . . . . 7 ⊢ (∅ ∈ ω → ((ω × {(inr‘∅)})‘∅) = (inr‘∅))
21	5, 20	ax-mp 5	. . . . . 6 ⊢ ((ω × {(inr‘∅)})‘∅) = (inr‘∅)
22	18, 21	eqtr4di 2247	. . . . 5 ⊢ (𝑦 ∈ (∅ ⊔ 1o) → 𝑦 = ((ω × {(inr‘∅)})‘∅))
23		fveq2 5561	. . . . . 6 ⊢ (𝑧 = ∅ → ((ω × {(inr‘∅)})‘𝑧) = ((ω × {(inr‘∅)})‘∅))
24	23	rspceeqv 2886	. . . . 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 2550	. . 3 ⊢ ∀𝑦 ∈ (∅ ⊔ 1o)∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧)
27		dffo3 5712	. . 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 4630	. . . 4 ⊢ ω ∈ V
30	19	snex 4219	. . . 4 ⊢ {(inr‘∅)} ∈ V
31	29, 30	xpex 4779	. . 3 ⊢ (ω × {(inr‘∅)}) ∈ V
32		foeq1 5479	. . 3 ⊢ (𝑓 = (ω × {(inr‘∅)}) → (𝑓:ω–onto→(∅ ⊔ 1o) ↔ (ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o)))
33	31, 32	spcev 2859	. 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 1506   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  ∅c0 3451  {csn 3623  ωcom 4627   × cxp 4662  ⟶wf 5255  –onto→wfo 5257  ‘cfv 5259  1_oc1o 6476   ⊔ cdju 7112  inlcinl 7120  inrcinr 7121

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-1st 6207  df-2nd 6208  df-1o 6483  df-dju 7113  df-inl 7122  df-inr 7123

This theorem is referenced by:  enumct  7190