Theorem en1 7076

Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.)

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 6691	. . . . 5 ⊢ 1o = {∅}
2	1	breq2i 4133	. . . 4 ⊢ (𝐴 ≈ 1o ↔ 𝐴 ≈ {∅})
3		bren 7020	. . . 4 ⊢ (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴–1-1-onto→{∅})
4	2, 3	bitri 184	. . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑓 𝑓:𝐴–1-1-onto→{∅})
5		f1ocnv 5647	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→{∅} → ◡𝑓:{∅}–1-1-onto→𝐴)
6		f1ofo 5641	. . . . . . . 8 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓:{∅}–onto→𝐴)
7		forn 5613	. . . . . . . 8 ⊢ (◡𝑓:{∅}–onto→𝐴 → ran ◡𝑓 = 𝐴)
8	6, 7	syl 14	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = 𝐴)
9		f1of 5634	. . . . . . . . . 10 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓:{∅}⟶𝐴)
10		0ex 4255	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
11	10	fsn2 5873	. . . . . . . . . . 11 ⊢ (◡𝑓:{∅}⟶𝐴 ↔ ((◡𝑓‘∅) ∈ 𝐴 ∧ ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩}))
12	11	simprbi 275	. . . . . . . . . 10 ⊢ (◡𝑓:{∅}⟶𝐴 → ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩})
13	9, 12	syl 14	. . . . . . . . 9 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩})
14	13	rneqd 5006	. . . . . . . 8 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = ran {⟨∅, (◡𝑓‘∅)⟩})
15	10	rnsnop 5263	. . . . . . . 8 ⊢ ran {⟨∅, (◡𝑓‘∅)⟩} = {(◡𝑓‘∅)}
16	14, 15	eqtrdi 2287	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = {(◡𝑓‘∅)})
17	8, 16	eqtr3d 2273	. . . . . 6 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → 𝐴 = {(◡𝑓‘∅)})
18	5, 17	syl 14	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→{∅} → 𝐴 = {(◡𝑓‘∅)})
19		f1ofn 5635	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓 Fn {∅})
20	10	snid 3736	. . . . . . 7 ⊢ ∅ ∈ {∅}
21		funfvex 5707	. . . . . . . 8 ⊢ ((Fun ◡𝑓 ∧ ∅ ∈ dom ◡𝑓) → (◡𝑓‘∅) ∈ V)
22	21	funfni 5478	. . . . . . 7 ⊢ ((◡𝑓 Fn {∅} ∧ ∅ ∈ {∅}) → (◡𝑓‘∅) ∈ V)
23	19, 20, 22	sylancl 417	. . . . . 6 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → (◡𝑓‘∅) ∈ V)
24		sneq 3716	. . . . . . . 8 ⊢ (𝑥 = (◡𝑓‘∅) → {𝑥} = {(◡𝑓‘∅)})
25	24	eqeq2d 2250	. . . . . . 7 ⊢ (𝑥 = (◡𝑓‘∅) → (𝐴 = {𝑥} ↔ 𝐴 = {(◡𝑓‘∅)}))
26	25	spcegv 2913	. . . . . 6 ⊢ ((◡𝑓‘∅) ∈ V → (𝐴 = {(◡𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥}))
27	23, 26	syl 14	. . . . 5 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → (𝐴 = {(◡𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥}))
28	5, 18, 27	sylc 62	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
29	28	exlimiv 1651	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
30	4, 29	sylbi 121	. 2 ⊢ (𝐴 ≈ 1o → ∃𝑥 𝐴 = {𝑥})
31		vex 2824	. . . . 5 ⊢ 𝑥 ∈ V
32	31	ensn1 7073	. . . 4 ⊢ {𝑥} ≈ 1o
33		breq1 4128	. . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o))
34	32, 33	mpbiri 168	. . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ≈ 1o)
35	34	exlimiv 1651	. 2 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1o)
36	30, 35	impbii 126	1 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821  ∅c0 3520  {csn 3705  ⟨cop 3708   class class class wbr 4125  ^◡ccnv 4768  ran crn 4770   Fn wfn 5367  ⟶wf 5368  –onto→wfo 5370  –1-1-onto→wf1o 5371  ‘cfv 5372  1_oc1o 6670   ≈ cen 7010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1o 6677  df-en 7013

This theorem is referenced by:  en1bg  7077  reuen1  7078  eqsndc  7200  pm54.43  7526  upgrex  16258  vtxdumgrfival  16453  1loopgrvd2fi  16460