Theorem en1 9062

Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.) Avoid ax-un 7753. (Revised by BTernaryTau, 23-Sep-2024.)

Assertion

Ref	Expression
en1	⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})

Distinct variable group:   𝑥,𝐴

Proof of Theorem en1

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df1o2 8511	. . . . 5 ⊢ 1o = {∅}
2	1	breq2i 5155	. . . 4 ⊢ (𝐴 ≈ 1o ↔ 𝐴 ≈ {∅})
3		encv 8991	. . . . . 6 ⊢ (𝐴 ≈ {∅} → (𝐴 ∈ V ∧ {∅} ∈ V))
4		breng 8992	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴–1-1-onto→{∅}))
5	3, 4	syl 17	. . . . 5 ⊢ (𝐴 ≈ {∅} → (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴–1-1-onto→{∅}))
6	5	ibi 267	. . . 4 ⊢ (𝐴 ≈ {∅} → ∃𝑓 𝑓:𝐴–1-1-onto→{∅})
7	2, 6	sylbi 217	. . 3 ⊢ (𝐴 ≈ 1o → ∃𝑓 𝑓:𝐴–1-1-onto→{∅})
8		f1ocnv 6860	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→{∅} → ◡𝑓:{∅}–1-1-onto→𝐴)
9		f1ofo 6855	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓:{∅}–onto→𝐴)
10		forn 6823	. . . . . . 7 ⊢ (◡𝑓:{∅}–onto→𝐴 → ran ◡𝑓 = 𝐴)
11	9, 10	syl 17	. . . . . 6 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = 𝐴)
12		f1of 6848	. . . . . . . . 9 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓:{∅}⟶𝐴)
13		0ex 5312	. . . . . . . . . . 11 ⊢ ∅ ∈ V
14	13	fsn2 7155	. . . . . . . . . 10 ⊢ (◡𝑓:{∅}⟶𝐴 ↔ ((◡𝑓‘∅) ∈ 𝐴 ∧ ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩}))
15	14	simprbi 496	. . . . . . . . 9 ⊢ (◡𝑓:{∅}⟶𝐴 → ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩})
16	12, 15	syl 17	. . . . . . . 8 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩})
17	16	rneqd 5951	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = ran {⟨∅, (◡𝑓‘∅)⟩})
18	13	rnsnop 6245	. . . . . . 7 ⊢ ran {⟨∅, (◡𝑓‘∅)⟩} = {(◡𝑓‘∅)}
19	17, 18	eqtrdi 2790	. . . . . 6 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = {(◡𝑓‘∅)})
20	11, 19	eqtr3d 2776	. . . . 5 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → 𝐴 = {(◡𝑓‘∅)})
21		fvex 6919	. . . . . 6 ⊢ (◡𝑓‘∅) ∈ V
22		sneq 4640	. . . . . . 7 ⊢ (𝑥 = (◡𝑓‘∅) → {𝑥} = {(◡𝑓‘∅)})
23	22	eqeq2d 2745	. . . . . 6 ⊢ (𝑥 = (◡𝑓‘∅) → (𝐴 = {𝑥} ↔ 𝐴 = {(◡𝑓‘∅)}))
24	21, 23	spcev 3605	. . . . 5 ⊢ (𝐴 = {(◡𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥})
25	8, 20, 24	3syl 18	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
26	25	exlimiv 1927	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
27	7, 26	syl 17	. 2 ⊢ (𝐴 ≈ 1o → ∃𝑥 𝐴 = {𝑥})
28		vex 3481	. . . . 5 ⊢ 𝑥 ∈ V
29	28	ensn1 9059	. . . 4 ⊢ {𝑥} ≈ 1o
30		breq1 5150	. . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o))
31	29, 30	mpbiri 258	. . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ≈ 1o)
32	31	exlimiv 1927	. 2 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1o)
33	27, 32	impbii 209	1 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1536  ∃wex 1775   ∈ wcel 2105  Vcvv 3477  ∅c0 4338  {csn 4630  ⟨cop 4636   class class class wbr 5147  ^◡ccnv 5687  ran crn 5689  ⟶wf 6558  –onto→wfo 6560  –1-1-onto→wf1o 6561  ‘cfv 6562  1_oc1o 8497   ≈ cen 8980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-1o 8504  df-en 8984

This theorem is referenced by:  en1b  9063  reuen1  9064  en1eqsn  9305  en2  9312  card1  10005  pm54.43  10038  hash1elsn  14406  hash1snb  14454  ufildom1  23949  unidifsnel  32560  unidifsnne  32561  funen1cnv  35080  lfuhgr3  35103  snen1g  43513