Theorem 0ct 7453

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 6717	. . . . 5 ⊢ ∅ ∈ 1o
2		djurcl 7398	. . . . 5 ⊢ (∅ ∈ 1o → (inr‘∅) ∈ (∅ ⊔ 1o))
3	1, 2	ax-mp 5	. . . 4 ⊢ (inr‘∅) ∈ (∅ ⊔ 1o)
4	3	fconst6 5595	. . 3 ⊢ (ω × {(inr‘∅)}):ω⟶(∅ ⊔ 1o)
5		peano1 4743	. . . . 5 ⊢ ∅ ∈ ω
6		rex0 3539	. . . . . . . . 9 ⊢ ¬ ∃𝑤 ∈ ∅ 𝑦 = (inl‘𝑤)
7		djur 7415	. . . . . . . . . . 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 736	. . . . . . . . 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 6705	. . . . . . . . 9 ⊢ 1o = {∅}
12	11	rexeqi 2754	. . . . . . . 8 ⊢ (∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤) ↔ ∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤))
13	10, 12	sylib 122	. . . . . . 7 ⊢ (𝑦 ∈ (∅ ⊔ 1o) → ∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤))
14		0ex 4262	. . . . . . . 8 ⊢ ∅ ∈ V
15		fveq2 5698	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (inr‘𝑤) = (inr‘∅))
16	15	eqeq2d 2250	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑦 = (inr‘𝑤) ↔ 𝑦 = (inr‘∅)))
17	14, 16	rexsn 3753	. . . . . . 7 ⊢ (∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤) ↔ 𝑦 = (inr‘∅))
18	13, 17	sylib 122	. . . . . 6 ⊢ (𝑦 ∈ (∅ ⊔ 1o) → 𝑦 = (inr‘∅))
19	3	elexi 2834	. . . . . . . 8 ⊢ (inr‘∅) ∈ V
20	19	fvconst2 5934	. . . . . . 7 ⊢ (∅ ∈ ω → ((ω × {(inr‘∅)})‘∅) = (inr‘∅))
21	5, 20	ax-mp 5	. . . . . 6 ⊢ ((ω × {(inr‘∅)})‘∅) = (inr‘∅)
22	18, 21	eqtr4di 2289	. . . . 5 ⊢ (𝑦 ∈ (∅ ⊔ 1o) → 𝑦 = ((ω × {(inr‘∅)})‘∅))
23		fveq2 5698	. . . . . 6 ⊢ (𝑧 = ∅ → ((ω × {(inr‘∅)})‘𝑧) = ((ω × {(inr‘∅)})‘∅))
24	23	rspceeqv 2948	. . . . 5 ⊢ ((∅ ∈ ω ∧ 𝑦 = ((ω × {(inr‘∅)})‘∅)) → ∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧))
25	5, 22, 24	sylancr 418	. . . 4 ⊢ (𝑦 ∈ (∅ ⊔ 1o) → ∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧))
26	25	rgen 2603	. . 3 ⊢ ∀𝑦 ∈ (∅ ⊔ 1o)∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧)
27		dffo3 5858	. . 3 ⊢ ((ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o) ↔ ((ω × {(inr‘∅)}):ω⟶(∅ ⊔ 1o) ∧ ∀𝑦 ∈ (∅ ⊔ 1o)∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧)))
28	4, 26, 27	mpbir2an 955	. 2 ⊢ (ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o)
29		omex 4742	. . . 4 ⊢ ω ∈ V
30	19	snex 4324	. . . 4 ⊢ {(inr‘∅)} ∈ V
31	29, 30	xpex 4893	. . 3 ⊢ (ω × {(inr‘∅)}) ∈ V
32		foeq1 5614	. . 3 ⊢ (𝑓 = (ω × {(inr‘∅)}) → (𝑓:ω–onto→(∅ ⊔ 1o) ↔ (ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o)))
33	31, 32	spcev 2920	. 2 ⊢ ((ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o) → ∃𝑓 𝑓:ω–onto→(∅ ⊔ 1o))
34	28, 33	ax-mp 5	1 ⊢ ∃𝑓 𝑓:ω–onto→(∅ ⊔ 1o)

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 720   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  ∅c0 3520  {csn 3709  ωcom 4739   × cxp 4774  ⟶wf 5376  –onto→wfo 5378  ‘cfv 5380  1_oc1o 6684   ⊔ cdju 7383  inlcinl 7391  inrcinr 7392

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-1st 6378  df-2nd 6379  df-1o 6691  df-dju 7384  df-inl 7393  df-inr 7394

This theorem is used by:  enumct  7461