Theorem dom1o 7045

Description: Two ways of saying that a set is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)

Assertion

Ref	Expression
dom1o	⊢ (𝐴 ∈ 𝑉 → (1o ≼ 𝐴 ↔ ∃𝑗 𝑗 ∈ 𝐴))

Distinct variable group:   𝐴,𝑗

Allowed substitution hint:   𝑉(𝑗)

Proof of Theorem dom1o

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdomg 6962	. . 3 ⊢ (𝐴 ∈ 𝑉 → (1o ≼ 𝐴 ↔ ∃𝑓 𝑓:1o–1-1→𝐴))
2		f1f 5551	. . . . . 6 ⊢ (𝑓:1o–1-1→𝐴 → 𝑓:1o⟶𝐴)
3		0lt1o 6651	. . . . . . 7 ⊢ ∅ ∈ 1o
4		ffvelcdm 5788	. . . . . . 7 ⊢ ((𝑓:1o⟶𝐴 ∧ ∅ ∈ 1o) → (𝑓‘∅) ∈ 𝐴)
5	3, 4	mpan2 425	. . . . . 6 ⊢ (𝑓:1o⟶𝐴 → (𝑓‘∅) ∈ 𝐴)
6		elex2 2820	. . . . . 6 ⊢ ((𝑓‘∅) ∈ 𝐴 → ∃𝑗 𝑗 ∈ 𝐴)
7	2, 5, 6	3syl 17	. . . . 5 ⊢ (𝑓:1o–1-1→𝐴 → ∃𝑗 𝑗 ∈ 𝐴)
8	7	a1i 9	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑓:1o–1-1→𝐴 → ∃𝑗 𝑗 ∈ 𝐴))
9	8	exlimdv 1867	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∃𝑓 𝑓:1o–1-1→𝐴 → ∃𝑗 𝑗 ∈ 𝐴))
10	1, 9	sylbid 150	. 2 ⊢ (𝐴 ∈ 𝑉 → (1o ≼ 𝐴 → ∃𝑗 𝑗 ∈ 𝐴))
11		0ex 4221	. . . . . . . 8 ⊢ ∅ ∈ V
12		vex 2806	. . . . . . . 8 ⊢ 𝑗 ∈ V
13	11, 12	opex 4327	. . . . . . 7 ⊢ ⟨∅, 𝑗⟩ ∈ V
14	13	snex 4281	. . . . . 6 ⊢ {⟨∅, 𝑗⟩} ∈ V
15	14	a1i 9	. . . . 5 ⊢ (𝑗 ∈ 𝐴 → {⟨∅, 𝑗⟩} ∈ V)
16		f1sng 5636	. . . . . . 7 ⊢ ((∅ ∈ 1o ∧ 𝑗 ∈ 𝐴) → {⟨∅, 𝑗⟩}:{∅}–1-1→𝐴)
17	3, 16	mpan 424	. . . . . 6 ⊢ (𝑗 ∈ 𝐴 → {⟨∅, 𝑗⟩}:{∅}–1-1→𝐴)
18		df1o2 6639	. . . . . . 7 ⊢ 1o = {∅}
19		f1eq2 5547	. . . . . . 7 ⊢ (1o = {∅} → ({⟨∅, 𝑗⟩}:1o–1-1→𝐴 ↔ {⟨∅, 𝑗⟩}:{∅}–1-1→𝐴))
20	18, 19	ax-mp 5	. . . . . 6 ⊢ ({⟨∅, 𝑗⟩}:1o–1-1→𝐴 ↔ {⟨∅, 𝑗⟩}:{∅}–1-1→𝐴)
21	17, 20	sylibr 134	. . . . 5 ⊢ (𝑗 ∈ 𝐴 → {⟨∅, 𝑗⟩}:1o–1-1→𝐴)
22		f1eq1 5546	. . . . 5 ⊢ (𝑓 = {⟨∅, 𝑗⟩} → (𝑓:1o–1-1→𝐴 ↔ {⟨∅, 𝑗⟩}:1o–1-1→𝐴))
23	15, 21, 22	elabd 2952	. . . 4 ⊢ (𝑗 ∈ 𝐴 → ∃𝑓 𝑓:1o–1-1→𝐴)
24	23	exlimiv 1647	. . 3 ⊢ (∃𝑗 𝑗 ∈ 𝐴 → ∃𝑓 𝑓:1o–1-1→𝐴)
25	24, 1	imbitrrid 156	. 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑗 𝑗 ∈ 𝐴 → 1o ≼ 𝐴))
26	10, 25	impbid 129	1 ⊢ (𝐴 ∈ 𝑉 → (1o ≼ 𝐴 ↔ ∃𝑗 𝑗 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2202  Vcvv 2803  ∅c0 3496  {csn 3673  ⟨cop 3676   class class class wbr 4093  ⟶wf 5329  –1-1→wf1 5330  ‘cfv 5333  1_oc1o 6618   ≼ cdom 6951

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1o 6625  df-dom 6954

This theorem is referenced by:  dom1oi  7046