Theorem en1 9089

Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.) Avoid ax-un 7783. (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 8521	. . . . 5 ⊢ 1o = {∅}
2	1	breq2i 5116	. . . 4 ⊢ (𝐴 ≈ 1o ↔ 𝐴 ≈ {∅})
3		encv 9019	. . . . . 6 ⊢ (𝐴 ≈ {∅} → (𝐴 ∈ V ∧ {∅} ∈ V))
4		breng 9020	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴⟶1-1onto{∅}))
5	3, 4	syl 18	. . . . 5 ⊢ (𝐴 ≈ {∅} → (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴⟶1-1onto{∅}))
6	5	ibi 270	. . . 4 ⊢ (𝐴 ≈ {∅} → ∃𝑓 𝑓:𝐴⟶1-1onto{∅})
7	2, 6	sylbi 220	. . 3 ⊢ (𝐴 ≈ 1o → ∃𝑓 𝑓:𝐴⟶1-1onto{∅})
8		f1ocnv 6867	. . . . 5 ⊢ (𝑓:𝐴⟶1-1onto{∅} → ◡𝑓:{∅}⟶1-1onto𝐴)
9		f1ofo 6862	. . . . . . 7 ⊢ (◡𝑓:{∅}⟶1-1onto𝐴 → ◡𝑓:{∅}⟶onto𝐴)
10		forn 6829	. . . . . . 7 ⊢ (◡𝑓:{∅}⟶onto𝐴 → ran ◡𝑓 = 𝐴)
11	9, 10	syl 18	. . . . . 6 ⊢ (◡𝑓:{∅}⟶1-1onto𝐴 → ran ◡𝑓 = 𝐴)
12		f1of 6854	. . . . . . . . 9 ⊢ (◡𝑓:{∅}⟶1-1onto𝐴 → ◡𝑓:{∅}⟶𝐴)
13		0ex 5267	. . . . . . . . . . 11 ⊢ ∅ ∈ V
14	13	fsn2 7169	. . . . . . . . . 10 ⊢ (◡𝑓:{∅}⟶𝐴 ↔ ((◡𝑓‘∅) ∈ 𝐴 ∧ ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩}))
15	14	simprbi 503	. . . . . . . . 9 ⊢ (◡𝑓:{∅}⟶𝐴 → ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩})
16	12, 15	syl 18	. . . . . . . 8 ⊢ (◡𝑓:{∅}⟶1-1onto𝐴 → ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩})
17	16	rneqd 5932	. . . . . . 7 ⊢ (◡𝑓:{∅}⟶1-1onto𝐴 → ran ◡𝑓 = ran {⟨∅, (◡𝑓‘∅)⟩})
18	13	rnsnop 6244	. . . . . . 7 ⊢ ran {⟨∅, (◡𝑓‘∅)⟩} = {(◡𝑓‘∅)}
19	17, 18	eqtrdi 2814	. . . . . 6 ⊢ (◡𝑓:{∅}⟶1-1onto𝐴 → ran ◡𝑓 = {(◡𝑓‘∅)})
20	11, 19	eqtr3d 2800	. . . . 5 ⊢ (◡𝑓:{∅}⟶1-1onto𝐴 → 𝐴 = {(◡𝑓‘∅)})
21		fvex 6929	. . . . . 6 ⊢ (◡𝑓‘∅) ∈ V
22		sneq 4594	. . . . . . 7 ⊢ (𝑥 = (◡𝑓‘∅) → {𝑥} = {(◡𝑓‘∅)})
23	22	eqeq2d 2774	. . . . . 6 ⊢ (𝑥 = (◡𝑓‘∅) → (𝐴 = {𝑥} ↔ 𝐴 = {(◡𝑓‘∅)}))
24	21, 23	spcev 3564	. . . . 5 ⊢ (𝐴 = {(◡𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥})
25	8, 20, 24	3syl 19	. . . 4 ⊢ (𝑓:𝐴⟶1-1onto{∅} → ∃𝑥 𝐴 = {𝑥})
26	25	exlimiv 1965	. . 3 ⊢ (∃𝑓 𝑓:𝐴⟶1-1onto{∅} → ∃𝑥 𝐴 = {𝑥})
27	7, 26	syl 18	. 2 ⊢ (𝐴 ≈ 1o → ∃𝑥 𝐴 = {𝑥})
28		vex 3458	. . . . 5 ⊢ 𝑥 ∈ V
29	28	ensn1 9086	. . . 4 ⊢ {𝑥} ≈ 1o
30		breq1 5111	. . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o))
31	29, 30	mpbiri 261	. . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ≈ 1o)
32	31	exlimiv 1965	. 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 2147  Vcvv 3454  ∅c0 4281  {csn 4588  ⟨cop 4840   class class class wbr 5108  ^◡ccnv 5657  ran crn 5659  ⟶wf 6562  ⟶ontowfo 6564  ⟶1-1ontowf1o 6565  ‘cfv 6566  1_oc1o 8507   ≈ cen 9008

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-1o 8514  df-en 9012

This theorem is used by:  en1b  9090  reuen1  9091  funen1cnv  9094  en1eqsn  9304  en2  9309  card1  10086  pm54.43  10121  hash1elsn  14556  hash1snb  14605  ufildom1  24300  lfuhgr3  29785  unidifsnel  33229  unidifsnne  33230  dflring3  34133  snen1g  44786  istermc3  50942