Theorem dom1o 7110

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 7026	. . 3 ⊢ (𝐴 ∈ 𝑉 → (1o ≼ 𝐴 ↔ ∃𝑓 𝑓:1o–1-1→𝐴))
2		f1f 5596	. . . . . 6 ⊢ (𝑓:1o–1-1→𝐴 → 𝑓:1o⟶𝐴)
3		0lt1o 6707	. . . . . . 7 ⊢ ∅ ∈ 1o
4		ffvelcdm 5835	. . . . . . 7 ⊢ ((𝑓:1o⟶𝐴 ∧ ∅ ∈ 1o) → (𝑓‘∅) ∈ 𝐴)
5	3, 4	mpan2 429	. . . . . 6 ⊢ (𝑓:1o⟶𝐴 → (𝑓‘∅) ∈ 𝐴)
6		elex2 2838	. . . . . 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 1872	. . 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 4258	. . . . . . . 8 ⊢ ∅ ∈ V
12		vex 2824	. . . . . . . 8 ⊢ 𝑗 ∈ V
13	11, 12	opex 4367	. . . . . . 7 ⊢ ⟨∅, 𝑗⟩ ∈ V
14	13	snex 4320	. . . . . 6 ⊢ {⟨∅, 𝑗⟩} ∈ V
15	14	a1i 9	. . . . 5 ⊢ (𝑗 ∈ 𝐴 → {⟨∅, 𝑗⟩} ∈ V)
16		f1sng 5681	. . . . . . 7 ⊢ ((∅ ∈ 1o ∧ 𝑗 ∈ 𝐴) → {⟨∅, 𝑗⟩}:{∅}–1-1→𝐴)
17	3, 16	mpan 428	. . . . . 6 ⊢ (𝑗 ∈ 𝐴 → {⟨∅, 𝑗⟩}:{∅}–1-1→𝐴)
18		df1o2 6695	. . . . . . 7 ⊢ 1o = {∅}
19		f1eq2 5592	. . . . . . 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 5591	. . . . 5 ⊢ (𝑓 = {⟨∅, 𝑗⟩} → (𝑓:1o–1-1→𝐴 ↔ {⟨∅, 𝑗⟩}:1o–1-1→𝐴))
23	15, 21, 22	elabd 2971	. . . 4 ⊢ (𝑗 ∈ 𝐴 → ∃𝑓 𝑓:1o–1-1→𝐴)
24	23	exlimiv 1651	. . 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 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821  ∅c0 3520  {csn 3708  ⟨cop 3711   class class class wbr 4128  ⟶wf 5371  –1-1→wf1 5372  ‘cfv 5375  1_oc1o 6674   ≼ cdom 7015

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 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 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-suc 4514  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-1o 6681  df-dom 7018

This theorem is referenced by:  dom1oi  7111