Theorem dom1o 6990

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 6910	. . 3 ⊢ (𝐴 ∈ 𝑉 → (1o ≼ 𝐴 ↔ ∃𝑓 𝑓:1o–1-1→𝐴))
2		f1f 5536	. . . . . 6 ⊢ (𝑓:1o–1-1→𝐴 → 𝑓:1o⟶𝐴)
3		0lt1o 6599	. . . . . . 7 ⊢ ∅ ∈ 1o
4		ffvelcdm 5773	. . . . . . 7 ⊢ ((𝑓:1o⟶𝐴 ∧ ∅ ∈ 1o) → (𝑓‘∅) ∈ 𝐴)
5	3, 4	mpan2 425	. . . . . 6 ⊢ (𝑓:1o⟶𝐴 → (𝑓‘∅) ∈ 𝐴)
6		elex2 2816	. . . . . 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 1865	. . 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 4211	. . . . . . . 8 ⊢ ∅ ∈ V
12		vex 2802	. . . . . . . 8 ⊢ 𝑗 ∈ V
13	11, 12	opex 4316	. . . . . . 7 ⊢ ⟨∅, 𝑗⟩ ∈ V
14	13	snex 4270	. . . . . 6 ⊢ {⟨∅, 𝑗⟩} ∈ V
15	14	a1i 9	. . . . 5 ⊢ (𝑗 ∈ 𝐴 → {⟨∅, 𝑗⟩} ∈ V)
16		f1sng 5620	. . . . . . 7 ⊢ ((∅ ∈ 1o ∧ 𝑗 ∈ 𝐴) → {⟨∅, 𝑗⟩}:{∅}–1-1→𝐴)
17	3, 16	mpan 424	. . . . . 6 ⊢ (𝑗 ∈ 𝐴 → {⟨∅, 𝑗⟩}:{∅}–1-1→𝐴)
18		df1o2 6587	. . . . . . 7 ⊢ 1o = {∅}
19		f1eq2 5532	. . . . . . 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 5531	. . . . 5 ⊢ (𝑓 = {⟨∅, 𝑗⟩} → (𝑓:1o–1-1→𝐴 ↔ {⟨∅, 𝑗⟩}:1o–1-1→𝐴))
23	15, 21, 22	elabd 2948	. . . 4 ⊢ (𝑗 ∈ 𝐴 → ∃𝑓 𝑓:1o–1-1→𝐴)
24	23	exlimiv 1644	. . 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 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2799  ∅c0 3491  {csn 3666  ⟨cop 3669   class class class wbr 4083  ⟶wf 5317  –1-1→wf1 5318  ‘cfv 5321  1_oc1o 6566   ≼ cdom 6899

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4385  df-suc 4463  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-1o 6573  df-dom 6902

This theorem is referenced by:  dom1oi  6991