Theorem en1 8999

Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.) Avoid ax-un 7713. (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 8438	. . . . 5 ⊢ 1o = {∅}
2	1	breq2i 5105	. . . 4 ⊢ (𝐴 ≈ 1o ↔ 𝐴 ≈ {∅})
3		encv 8929	. . . . . 6 ⊢ (𝐴 ≈ {∅} → (𝐴 ∈ V ∧ {∅} ∈ V))
4		breng 8930	. . . . . 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 269	. . . 4 ⊢ (𝐴 ≈ {∅} → ∃𝑓 𝑓:𝐴–1-1-onto→{∅})
7	2, 6	sylbi 219	. . 3 ⊢ (𝐴 ≈ 1o → ∃𝑓 𝑓:𝐴–1-1-onto→{∅})
8		f1ocnv 6814	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→{∅} → ◡𝑓:{∅}–1-1-onto→𝐴)
9		f1ofo 6809	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓:{∅}–onto→𝐴)
10		forn 6776	. . . . . . 7 ⊢ (◡𝑓:{∅}–onto→𝐴 → ran ◡𝑓 = 𝐴)
11	9, 10	syl 17	. . . . . 6 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = 𝐴)
12		f1of 6801	. . . . . . . . 9 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓:{∅}⟶𝐴)
13		0ex 5254	. . . . . . . . . . 11 ⊢ ∅ ∈ V
14	13	fsn2 7113	. . . . . . . . . 10 ⊢ (◡𝑓:{∅}⟶𝐴 ↔ ((◡𝑓‘∅) ∈ 𝐴 ∧ ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩}))
15	14	simprbi 501	. . . . . . . . 9 ⊢ (◡𝑓:{∅}⟶𝐴 → ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩})
16	12, 15	syl 17	. . . . . . . 8 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩})
17	16	rneqd 5910	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = ran {⟨∅, (◡𝑓‘∅)⟩})
18	13	rnsnop 6206	. . . . . . 7 ⊢ ran {⟨∅, (◡𝑓‘∅)⟩} = {(◡𝑓‘∅)}
19	17, 18	eqtrdi 2812	. . . . . 6 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = {(◡𝑓‘∅)})
20	11, 19	eqtr3d 2798	. . . . 5 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → 𝐴 = {(◡𝑓‘∅)})
21		fvex 6875	. . . . . 6 ⊢ (◡𝑓‘∅) ∈ V
22		sneq 4589	. . . . . . 7 ⊢ (𝑥 = (◡𝑓‘∅) → {𝑥} = {(◡𝑓‘∅)})
23	22	eqeq2d 2772	. . . . . 6 ⊢ (𝑥 = (◡𝑓‘∅) → (𝐴 = {𝑥} ↔ 𝐴 = {(◡𝑓‘∅)}))
24	21, 23	spcev 3564	. . . . 5 ⊢ (𝐴 = {(◡𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥})
25	8, 20, 24	3syl 18	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
26	25	exlimiv 1949	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
27	7, 26	syl 17	. 2 ⊢ (𝐴 ≈ 1o → ∃𝑥 𝐴 = {𝑥})
28		vex 3457	. . . . 5 ⊢ 𝑥 ∈ V
29	28	ensn1 8996	. . . 4 ⊢ {𝑥} ≈ 1o
30		breq1 5100	. . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o))
31	29, 30	mpbiri 260	. . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ≈ 1o)
32	31	exlimiv 1949	. 2 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1o)
33	27, 32	impbii 211	1 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  Vcvv 3453  ∅c0 4283  {csn 4579  ⟨cop 4585   class class class wbr 5097  ^◡ccnv 5642  ran crn 5644  ⟶wf 6512  –onto→wfo 6514  –1-1-onto→wf1o 6515  ‘cfv 6516  1_oc1o 8424   ≈ cen 8918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-1o 8431  df-en 8922

This theorem is referenced by:  en1b  9000  reuen1  9001  en1eqsn  9213  en2  9218  card1  9920  pm54.43  9953  hash1elsn  14378  hash1snb  14426  ufildom1  23974  unidifsnel  32694  unidifsnne  32695  dflring3  33654  funen1cnv  35343  lfuhgr3  35431  snen1g  44061  istermc3  50058