Theorem en1 9023

Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.) Avoid ax-un 7738. (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 8462	. . . . 5 ⊢ 1o = {∅}
2	1	breq2i 5119	. . . 4 ⊢ (𝐴 ≈ 1o ↔ 𝐴 ≈ {∅})
3		encv 8953	. . . . . 6 ⊢ (𝐴 ≈ {∅} → (𝐴 ∈ V ∧ {∅} ∈ V))
4		breng 8954	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴–1-1-onto→{∅}))
5	3, 4	syl 18	. . . . 5 ⊢ (𝐴 ≈ {∅} → (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴–1-1-onto→{∅}))
6	5	ibi 270	. . . 4 ⊢ (𝐴 ≈ {∅} → ∃𝑓 𝑓:𝐴–1-1-onto→{∅})
7	2, 6	sylbi 220	. . 3 ⊢ (𝐴 ≈ 1o → ∃𝑓 𝑓:𝐴–1-1-onto→{∅})
8		f1ocnv 6837	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→{∅} → ◡𝑓:{∅}–1-1-onto→𝐴)
9		f1ofo 6832	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓:{∅}–onto→𝐴)
10		forn 6799	. . . . . . 7 ⊢ (◡𝑓:{∅}–onto→𝐴 → ran ◡𝑓 = 𝐴)
11	9, 10	syl 18	. . . . . 6 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = 𝐴)
12		f1of 6824	. . . . . . . . 9 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓:{∅}⟶𝐴)
13		0ex 5272	. . . . . . . . . . 11 ⊢ ∅ ∈ V
14	13	fsn2 7136	. . . . . . . . . 10 ⊢ (◡𝑓:{∅}⟶𝐴 ↔ ((◡𝑓‘∅) ∈ 𝐴 ∧ ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩}))
15	14	simprbi 503	. . . . . . . . 9 ⊢ (◡𝑓:{∅}⟶𝐴 → ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩})
16	12, 15	syl 18	. . . . . . . 8 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩})
17	16	rneqd 5930	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = ran {⟨∅, (◡𝑓‘∅)⟩})
18	13	rnsnop 6227	. . . . . . 7 ⊢ ran {⟨∅, (◡𝑓‘∅)⟩} = {(◡𝑓‘∅)}
19	17, 18	eqtrdi 2816	. . . . . 6 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = {(◡𝑓‘∅)})
20	11, 19	eqtr3d 2802	. . . . 5 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → 𝐴 = {(◡𝑓‘∅)})
21		fvex 6898	. . . . . 6 ⊢ (◡𝑓‘∅) ∈ V
22		sneq 4601	. . . . . . 7 ⊢ (𝑥 = (◡𝑓‘∅) → {𝑥} = {(◡𝑓‘∅)})
23	22	eqeq2d 2776	. . . . . 6 ⊢ (𝑥 = (◡𝑓‘∅) → (𝐴 = {𝑥} ↔ 𝐴 = {(◡𝑓‘∅)}))
24	21, 23	spcev 3567	. . . . 5 ⊢ (𝐴 = {(◡𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥})
25	8, 20, 24	3syl 19	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
26	25	exlimiv 1963	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
27	7, 26	syl 18	. 2 ⊢ (𝐴 ≈ 1o → ∃𝑥 𝐴 = {𝑥})
28		vex 3461	. . . . 5 ⊢ 𝑥 ∈ V
29	28	ensn1 9020	. . . 4 ⊢ {𝑥} ≈ 1o
30		breq1 5114	. . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o))
31	29, 30	mpbiri 261	. . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ≈ 1o)
32	31	exlimiv 1963	. 2 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1o)
33	27, 32	impbii 212	1 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  Vcvv 3457  ∅c0 4286  {csn 4591  ⟨cop 4597   class class class wbr 5111  ^◡ccnv 5662  ran crn 5664  ⟶wf 6536  –onto→wfo 6538  –1-1-onto→wf1o 6539  ‘cfv 6540  1_oc1o 8448   ≈ cen 8942

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-1o 8455  df-en 8946

This theorem is used by:  en1b  9024  reuen1  9025  funen1cnv  9028  en1eqsn  9238  en2  9243  card1  9966  pm54.43  9999  hash1elsn  14421  hash1snb  14470  ufildom1  24114  lfuhgr3  29531  unidifsnel  32928  unidifsnne  32929  dflring3  33827  snen1g  44283  istermc3  50287