Theorem en1 6949

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 6573	. . . . 5 ⊢ 1o = {∅}
2	1	breq2i 4090	. . . 4 ⊢ (𝐴 ≈ 1o ↔ 𝐴 ≈ {∅})
3		bren 6893	. . . 4 ⊢ (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴–1-1-onto→{∅})
4	2, 3	bitri 184	. . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑓 𝑓:𝐴–1-1-onto→{∅})
5		f1ocnv 5584	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→{∅} → ◡𝑓:{∅}–1-1-onto→𝐴)
6		f1ofo 5578	. . . . . . . 8 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓:{∅}–onto→𝐴)
7		forn 5550	. . . . . . . 8 ⊢ (◡𝑓:{∅}–onto→𝐴 → ran ◡𝑓 = 𝐴)
8	6, 7	syl 14	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = 𝐴)
9		f1of 5571	. . . . . . . . . 10 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓:{∅}⟶𝐴)
10		0ex 4210	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
11	10	fsn2 5808	. . . . . . . . . . 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 4952	. . . . . . . 8 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = ran {⟨∅, (◡𝑓‘∅)⟩})
15	10	rnsnop 5208	. . . . . . . 8 ⊢ ran {⟨∅, (◡𝑓‘∅)⟩} = {(◡𝑓‘∅)}
16	14, 15	eqtrdi 2278	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = {(◡𝑓‘∅)})
17	8, 16	eqtr3d 2264	. . . . . 6 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → 𝐴 = {(◡𝑓‘∅)})
18	5, 17	syl 14	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→{∅} → 𝐴 = {(◡𝑓‘∅)})
19		f1ofn 5572	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓 Fn {∅})
20	10	snid 3697	. . . . . . 7 ⊢ ∅ ∈ {∅}
21		funfvex 5643	. . . . . . . 8 ⊢ ((Fun ◡𝑓 ∧ ∅ ∈ dom ◡𝑓) → (◡𝑓‘∅) ∈ V)
22	21	funfni 5422	. . . . . . 7 ⊢ ((◡𝑓 Fn {∅} ∧ ∅ ∈ {∅}) → (◡𝑓‘∅) ∈ V)
23	19, 20, 22	sylancl 413	. . . . . 6 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → (◡𝑓‘∅) ∈ V)
24		sneq 3677	. . . . . . . 8 ⊢ (𝑥 = (◡𝑓‘∅) → {𝑥} = {(◡𝑓‘∅)})
25	24	eqeq2d 2241	. . . . . . 7 ⊢ (𝑥 = (◡𝑓‘∅) → (𝐴 = {𝑥} ↔ 𝐴 = {(◡𝑓‘∅)}))
26	25	spcegv 2891	. . . . . 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 1644	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
30	4, 29	sylbi 121	. 2 ⊢ (𝐴 ≈ 1o → ∃𝑥 𝐴 = {𝑥})
31		vex 2802	. . . . 5 ⊢ 𝑥 ∈ V
32	31	ensn1 6946	. . . 4 ⊢ {𝑥} ≈ 1o
33		breq1 4085	. . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o))
34	32, 33	mpbiri 168	. . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ≈ 1o)
35	34	exlimiv 1644	. 2 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1o)
36	30, 35	impbii 126	1 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2799  ∅c0 3491  {csn 3666  ⟨cop 3669   class class class wbr 4082  ^◡ccnv 4717  ran crn 4719   Fn wfn 5312  ⟶wf 5313  –onto→wfo 5315  –1-1-onto→wf1o 5316  ‘cfv 5317  1_oc1o 6553   ≈ cen 6883

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-suc 4461  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-1o 6560  df-en 6886

This theorem is referenced by:  en1bg  6950  reuen1  6951  pm54.43  7359  upgrex  15897