Theorem 0ct 7224

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 6539	. . . . 5 ⊢ ∅ ∈ 1o
2		djurcl 7169	. . . . 5 ⊢ (∅ ∈ 1o → (inr‘∅) ∈ (∅ ⊔ 1o))
3	1, 2	ax-mp 5	. . . 4 ⊢ (inr‘∅) ∈ (∅ ⊔ 1o)
4	3	fconst6 5487	. . 3 ⊢ (ω × {(inr‘∅)}):ω⟶(∅ ⊔ 1o)
5		peano1 4650	. . . . 5 ⊢ ∅ ∈ ω
6		rex0 3482	. . . . . . . . 9 ⊢ ¬ ∃𝑤 ∈ ∅ 𝑦 = (inl‘𝑤)
7		djur 7186	. . . . . . . . . . 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 726	. . . . . . . . 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 6528	. . . . . . . . 9 ⊢ 1o = {∅}
12	11	rexeqi 2708	. . . . . . . 8 ⊢ (∃𝑤 ∈ 1o 𝑦 = (inr‘𝑤) ↔ ∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤))
13	10, 12	sylib 122	. . . . . . 7 ⊢ (𝑦 ∈ (∅ ⊔ 1o) → ∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤))
14		0ex 4179	. . . . . . . 8 ⊢ ∅ ∈ V
15		fveq2 5589	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (inr‘𝑤) = (inr‘∅))
16	15	eqeq2d 2218	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑦 = (inr‘𝑤) ↔ 𝑦 = (inr‘∅)))
17	14, 16	rexsn 3682	. . . . . . 7 ⊢ (∃𝑤 ∈ {∅}𝑦 = (inr‘𝑤) ↔ 𝑦 = (inr‘∅))
18	13, 17	sylib 122	. . . . . 6 ⊢ (𝑦 ∈ (∅ ⊔ 1o) → 𝑦 = (inr‘∅))
19	3	elexi 2786	. . . . . . . 8 ⊢ (inr‘∅) ∈ V
20	19	fvconst2 5813	. . . . . . 7 ⊢ (∅ ∈ ω → ((ω × {(inr‘∅)})‘∅) = (inr‘∅))
21	5, 20	ax-mp 5	. . . . . 6 ⊢ ((ω × {(inr‘∅)})‘∅) = (inr‘∅)
22	18, 21	eqtr4di 2257	. . . . 5 ⊢ (𝑦 ∈ (∅ ⊔ 1o) → 𝑦 = ((ω × {(inr‘∅)})‘∅))
23		fveq2 5589	. . . . . 6 ⊢ (𝑧 = ∅ → ((ω × {(inr‘∅)})‘𝑧) = ((ω × {(inr‘∅)})‘∅))
24	23	rspceeqv 2899	. . . . 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 2560	. . 3 ⊢ ∀𝑦 ∈ (∅ ⊔ 1o)∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧)
27		dffo3 5740	. . 3 ⊢ ((ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o) ↔ ((ω × {(inr‘∅)}):ω⟶(∅ ⊔ 1o) ∧ ∀𝑦 ∈ (∅ ⊔ 1o)∃𝑧 ∈ ω 𝑦 = ((ω × {(inr‘∅)})‘𝑧)))
28	4, 26, 27	mpbir2an 945	. 2 ⊢ (ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o)
29		omex 4649	. . . 4 ⊢ ω ∈ V
30	19	snex 4237	. . . 4 ⊢ {(inr‘∅)} ∈ V
31	29, 30	xpex 4798	. . 3 ⊢ (ω × {(inr‘∅)}) ∈ V
32		foeq1 5506	. . 3 ⊢ (𝑓 = (ω × {(inr‘∅)}) → (𝑓:ω–onto→(∅ ⊔ 1o) ↔ (ω × {(inr‘∅)}):ω–onto→(∅ ⊔ 1o)))
33	31, 32	spcev 2872	. 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 710   = wceq 1373  ∃wex 1516   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486  ∅c0 3464  {csn 3638  ωcom 4646   × cxp 4681  ⟶wf 5276  –onto→wfo 5278  ‘cfv 5280  1_oc1o 6508   ⊔ cdju 7154  inlcinl 7162  inrcinr 7163

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-iord 4421  df-on 4423  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-1st 6239  df-2nd 6240  df-1o 6515  df-dju 7155  df-inl 7164  df-inr 7165

This theorem is referenced by:  enumct  7232