Theorem dom1o 16043

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 6847	. . 3 ⊢ (𝐴 ∈ 𝑉 → (1o ≼ 𝐴 ↔ ∃𝑓 𝑓:1o–1-1→𝐴))
2		f1f 5490	. . . . . 6 ⊢ (𝑓:1o–1-1→𝐴 → 𝑓:1o⟶𝐴)
3		0lt1o 6536	. . . . . . 7 ⊢ ∅ ∈ 1o
4		ffvelcdm 5723	. . . . . . 7 ⊢ ((𝑓:1o⟶𝐴 ∧ ∅ ∈ 1o) → (𝑓‘∅) ∈ 𝐴)
5	3, 4	mpan2 425	. . . . . 6 ⊢ (𝑓:1o⟶𝐴 → (𝑓‘∅) ∈ 𝐴)
6		elex2 2790	. . . . . 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 1843	. . 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 4176	. . . . . . . 8 ⊢ ∅ ∈ V
12		vex 2776	. . . . . . . 8 ⊢ 𝑗 ∈ V
13	11, 12	opex 4278	. . . . . . 7 ⊢ ⟨∅, 𝑗⟩ ∈ V
14	13	snex 4234	. . . . . 6 ⊢ {⟨∅, 𝑗⟩} ∈ V
15	14	a1i 9	. . . . 5 ⊢ (𝑗 ∈ 𝐴 → {⟨∅, 𝑗⟩} ∈ V)
16		f1sng 5574	. . . . . . 7 ⊢ ((∅ ∈ 1o ∧ 𝑗 ∈ 𝐴) → {⟨∅, 𝑗⟩}:{∅}–1-1→𝐴)
17	3, 16	mpan 424	. . . . . 6 ⊢ (𝑗 ∈ 𝐴 → {⟨∅, 𝑗⟩}:{∅}–1-1→𝐴)
18		df1o2 6525	. . . . . . 7 ⊢ 1o = {∅}
19		f1eq2 5486	. . . . . . 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 5485	. . . . 5 ⊢ (𝑓 = {⟨∅, 𝑗⟩} → (𝑓:1o–1-1→𝐴 ↔ {⟨∅, 𝑗⟩}:1o–1-1→𝐴))
23	15, 21, 22	elabd 2920	. . . 4 ⊢ (𝑗 ∈ 𝐴 → ∃𝑓 𝑓:1o–1-1→𝐴)
24	23	exlimiv 1622	. . 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 1373  ∃wex 1516   ∈ wcel 2177  Vcvv 2773  ∅c0 3462  {csn 3635  ⟨cop 3638   class class class wbr 4048  ⟶wf 5273  –1-1→wf1 5274  ‘cfv 5277  1_oc1o 6505   ≼ cdom 6836

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 4167  ax-nul 4175  ax-pow 4223  ax-pr 4258  ax-un 4485

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  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 3001  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-br 4049  df-opab 4111  df-id 4345  df-suc 4423  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-1o 6512  df-dom 6839

This theorem is referenced by: (None)