Theorem en1 6279

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 6013	. . . . 5 ⊢ 1𝑜 = {∅}
2	1	breq2i 3772	. . . 4 ⊢ (𝐴 ≈ 1𝑜 ↔ 𝐴 ≈ {∅})
3		bren 6228	. . . 4 ⊢ (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴–1-1-onto→{∅})
4	2, 3	bitri 173	. . 3 ⊢ (𝐴 ≈ 1𝑜 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→{∅})
5		f1ocnv 5139	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→{∅} → ◡𝑓:{∅}–1-1-onto→𝐴)
6		f1ofo 5133	. . . . . . . 8 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓:{∅}–onto→𝐴)
7		forn 5109	. . . . . . . 8 ⊢ (◡𝑓:{∅}–onto→𝐴 → ran ◡𝑓 = 𝐴)
8	6, 7	syl 14	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = 𝐴)
9		f1of 5126	. . . . . . . . . 10 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓:{∅}⟶𝐴)
10		0ex 3884	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
11	10	fsn2 5337	. . . . . . . . . . 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 4563	. . . . . . . 8 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = ran {⟨∅, (◡𝑓‘∅)⟩})
15	10	rnsnop 4801	. . . . . . . 8 ⊢ ran {⟨∅, (◡𝑓‘∅)⟩} = {(◡𝑓‘∅)}
16	14, 15	syl6eq 2088	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = {(◡𝑓‘∅)})
17	8, 16	eqtr3d 2074	. . . . . 6 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → 𝐴 = {(◡𝑓‘∅)})
18	5, 17	syl 14	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→{∅} → 𝐴 = {(◡𝑓‘∅)})
19		f1ofn 5127	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓 Fn {∅})
20	10	snid 3402	. . . . . . 7 ⊢ ∅ ∈ {∅}
21		funfvex 5192	. . . . . . . 8 ⊢ ((Fun ◡𝑓 ∧ ∅ ∈ dom ◡𝑓) → (◡𝑓‘∅) ∈ V)
22	21	funfni 4999	. . . . . . 7 ⊢ ((◡𝑓 Fn {∅} ∧ ∅ ∈ {∅}) → (◡𝑓‘∅) ∈ V)
23	19, 20, 22	sylancl 392	. . . . . 6 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → (◡𝑓‘∅) ∈ V)
24		sneq 3386	. . . . . . . 8 ⊢ (𝑥 = (◡𝑓‘∅) → {𝑥} = {(◡𝑓‘∅)})
25	24	eqeq2d 2051	. . . . . . 7 ⊢ (𝑥 = (◡𝑓‘∅) → (𝐴 = {𝑥} ↔ 𝐴 = {(◡𝑓‘∅)}))
26	25	spcegv 2641	. . . . . 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 1489	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
30	4, 29	sylbi 114	. 2 ⊢ (𝐴 ≈ 1𝑜 → ∃𝑥 𝐴 = {𝑥})
31		vex 2560	. . . . 5 ⊢ 𝑥 ∈ V
32	31	ensn1 6276	. . . 4 ⊢ {𝑥} ≈ 1𝑜
33		breq1 3767	. . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ≈ 1𝑜 ↔ {𝑥} ≈ 1𝑜))
34	32, 33	mpbiri 157	. . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ≈ 1𝑜)
35	34	exlimiv 1489	. 2 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1𝑜)
36	30, 35	impbii 117	1 ⊢ (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557  ∅c0 3224  {csn 3375  ⟨cop 3378   class class class wbr 3764  ^◡ccnv 4344  ran crn 4346   Fn wfn 4897  ⟶wf 4898  –onto→wfo 4900  –1-1-onto→wf1o 4901  ‘cfv 4902  1_𝑜c1o 5994   ≈ cen 6219

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-1o 6001  df-en 6222

This theorem is referenced by:  en1bg  6280  reuen1  6281