Theorem en1 6215

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 5952	. . . . 5 ⊢ 1𝑜 = {∅}
2	1	breq2i 3763	. . . 4 ⊢ (A ≈ 1𝑜 ↔ A ≈ {∅})
3		bren 6164	. . . 4 ⊢ (A ≈ {∅} ↔ ∃f f:A–1-1-onto→{∅})
4	2, 3	bitri 173	. . 3 ⊢ (A ≈ 1𝑜 ↔ ∃f f:A–1-1-onto→{∅})
5		f1ocnv 5082	. . . . 5 ⊢ (f:A–1-1-onto→{∅} → ◡f:{∅}–1-1-onto→A)
6		f1ofo 5076	. . . . . . . 8 ⊢ (◡f:{∅}–1-1-onto→A → ◡f:{∅}–onto→A)
7		forn 5052	. . . . . . . 8 ⊢ (◡f:{∅}–onto→A → ran ◡f = A)
8	6, 7	syl 14	. . . . . . 7 ⊢ (◡f:{∅}–1-1-onto→A → ran ◡f = A)
9		f1of 5069	. . . . . . . . . 10 ⊢ (◡f:{∅}–1-1-onto→A → ◡f:{∅}⟶A)
10		0ex 3875	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
11	10	fsn2 5280	. . . . . . . . . . 11 ⊢ (◡f:{∅}⟶A ↔ ((◡f‘∅) ∈ A ∧ ◡f = {⟨∅, (◡f‘∅)⟩}))
12	11	simprbi 260	. . . . . . . . . 10 ⊢ (◡f:{∅}⟶A → ◡f = {⟨∅, (◡f‘∅)⟩})
13	9, 12	syl 14	. . . . . . . . 9 ⊢ (◡f:{∅}–1-1-onto→A → ◡f = {⟨∅, (◡f‘∅)⟩})
14	13	rneqd 4506	. . . . . . . 8 ⊢ (◡f:{∅}–1-1-onto→A → ran ◡f = ran {⟨∅, (◡f‘∅)⟩})
15	10	rnsnop 4744	. . . . . . . 8 ⊢ ran {⟨∅, (◡f‘∅)⟩} = {(◡f‘∅)}
16	14, 15	syl6eq 2085	. . . . . . 7 ⊢ (◡f:{∅}–1-1-onto→A → ran ◡f = {(◡f‘∅)})
17	8, 16	eqtr3d 2071	. . . . . 6 ⊢ (◡f:{∅}–1-1-onto→A → A = {(◡f‘∅)})
18	5, 17	syl 14	. . . . 5 ⊢ (f:A–1-1-onto→{∅} → A = {(◡f‘∅)})
19		f1ofn 5070	. . . . . . 7 ⊢ (◡f:{∅}–1-1-onto→A → ◡f Fn {∅})
20	10	snid 3394	. . . . . . 7 ⊢ ∅ ∈ {∅}
21		funfvex 5135	. . . . . . . 8 ⊢ ((Fun ◡f ∧ ∅ ∈ dom ◡f) → (◡f‘∅) ∈ V)
22	21	funfni 4942	. . . . . . 7 ⊢ ((◡f Fn {∅} ∧ ∅ ∈ {∅}) → (◡f‘∅) ∈ V)
23	19, 20, 22	sylancl 392	. . . . . 6 ⊢ (◡f:{∅}–1-1-onto→A → (◡f‘∅) ∈ V)
24		sneq 3378	. . . . . . . 8 ⊢ (x = (◡f‘∅) → {x} = {(◡f‘∅)})
25	24	eqeq2d 2048	. . . . . . 7 ⊢ (x = (◡f‘∅) → (A = {x} ↔ A = {(◡f‘∅)}))
26	25	spcegv 2635	. . . . . 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 56	. . . 4 ⊢ (f:A–1-1-onto→{∅} → ∃x A = {x})
29	28	exlimiv 1486	. . 3 ⊢ (∃f f:A–1-1-onto→{∅} → ∃x A = {x})
30	4, 29	sylbi 114	. 2 ⊢ (A ≈ 1𝑜 → ∃x A = {x})
31		vex 2554	. . . . 5 ⊢ x ∈ V
32	31	ensn1 6212	. . . 4 ⊢ {x} ≈ 1𝑜
33		breq1 3758	. . . 4 ⊢ (A = {x} → (A ≈ 1𝑜 ↔ {x} ≈ 1𝑜))
34	32, 33	mpbiri 157	. . 3 ⊢ (A = {x} → A ≈ 1𝑜)
35	34	exlimiv 1486	. 2 ⊢ (∃x A = {x} → A ≈ 1𝑜)
36	30, 35	impbii 117	1 ⊢ (A ≈ 1𝑜 ↔ ∃x A = {x})

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551  ∅c0 3218  {csn 3367  ⟨cop 3370   class class class wbr 3755  ^◡ccnv 4287  ran crn 4289   Fn wfn 4840  ⟶wf 4841  –onto→wfo 4843  –1-1-onto→wf1o 4844  ‘cfv 4845  1_𝑜c1o 5933   ≈ cen 6155

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-1o 5940  df-en 6158

This theorem is referenced by:  en1bg  6216  reuen1  6217