Theorem en1 9017

Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.) Avoid ax-un 7732. (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 8456	. . . . 5 ⊢ 1o = {∅}
2	1	breq2i 5117	. . . 4 ⊢ (𝐴 ≈ 1o ↔ 𝐴 ≈ {∅})
3		encv 8947	. . . . . 6 ⊢ (𝐴 ≈ {∅} → (𝐴 ∈ V ∧ {∅} ∈ V))
4		breng 8948	. . . . . 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 6833	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→{∅} → ◡𝑓:{∅}–1-1-onto→𝐴)
9		f1ofo 6828	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓:{∅}–onto→𝐴)
10		forn 6795	. . . . . . 7 ⊢ (◡𝑓:{∅}–onto→𝐴 → ran ◡𝑓 = 𝐴)
11	9, 10	syl 18	. . . . . 6 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = 𝐴)
12		f1of 6820	. . . . . . . . 9 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓:{∅}⟶𝐴)
13		0ex 5270	. . . . . . . . . . 11 ⊢ ∅ ∈ V
14	13	fsn2 7132	. . . . . . . . . 10 ⊢ (◡𝑓:{∅}⟶𝐴 ↔ ((◡𝑓‘∅) ∈ 𝐴 ∧ ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩}))
15	14	simprbi 502	. . . . . . . . 9 ⊢ (◡𝑓:{∅}⟶𝐴 → ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩})
16	12, 15	syl 18	. . . . . . . 8 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩})
17	16	rneqd 5928	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = ran {⟨∅, (◡𝑓‘∅)⟩})
18	13	rnsnop 6225	. . . . . . 7 ⊢ ran {⟨∅, (◡𝑓‘∅)⟩} = {(◡𝑓‘∅)}
19	17, 18	eqtrdi 2814	. . . . . 6 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = {(◡𝑓‘∅)})
20	11, 19	eqtr3d 2800	. . . . 5 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → 𝐴 = {(◡𝑓‘∅)})
21		fvex 6894	. . . . . 6 ⊢ (◡𝑓‘∅) ∈ V
22		sneq 4599	. . . . . . 7 ⊢ (𝑥 = (◡𝑓‘∅) → {𝑥} = {(◡𝑓‘∅)})
23	22	eqeq2d 2774	. . . . . 6 ⊢ (𝑥 = (◡𝑓‘∅) → (𝐴 = {𝑥} ↔ 𝐴 = {(◡𝑓‘∅)}))
24	21, 23	spcev 3565	. . . . 5 ⊢ (𝐴 = {(◡𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥})
25	8, 20, 24	3syl 19	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
26	25	exlimiv 1960	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
27	7, 26	syl 18	. 2 ⊢ (𝐴 ≈ 1o → ∃𝑥 𝐴 = {𝑥})
28		vex 3459	. . . . 5 ⊢ 𝑥 ∈ V
29	28	ensn1 9014	. . . 4 ⊢ {𝑥} ≈ 1o
30		breq1 5112	. . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o))
31	29, 30	mpbiri 261	. . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ≈ 1o)
32	31	exlimiv 1960	. 2 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1o)
33	27, 32	impbii 212	1 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455  ∅c0 4286  {csn 4589  ⟨cop 4595   class class class wbr 5109  ^◡ccnv 5660  ran crn 5662  ⟶wf 6532  –onto→wfo 6534  –1-1-onto→wf1o 6535  ‘cfv 6536  1_oc1o 8442   ≈ cen 8936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-1o 8449  df-en 8940

This theorem is referenced by:  en1b  9018  reuen1  9019  en1eqsn  9231  en2  9236  card1  9950  pm54.43  9983  hash1elsn  14403  hash1snb  14452  ufildom1  24083  unidifsnel  32881  unidifsnne  32882  dflring3  33787  funen1cnv  35477  lfuhgr3  35612  snen1g  44250  istermc3  50254