Theorem en1 8579

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 8119	. . . . 5 ⊢ 1o = {∅}
2	1	breq2i 5077	. . . 4 ⊢ (𝐴 ≈ 1o ↔ 𝐴 ≈ {∅})
3		bren 8521	. . . 4 ⊢ (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴–1-1-onto→{∅})
4	2, 3	bitri 277	. . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑓 𝑓:𝐴–1-1-onto→{∅})
5		f1ocnv 6630	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→{∅} → ◡𝑓:{∅}–1-1-onto→𝐴)
6		f1ofo 6625	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓:{∅}–onto→𝐴)
7		forn 6596	. . . . . . 7 ⊢ (◡𝑓:{∅}–onto→𝐴 → ran ◡𝑓 = 𝐴)
8	6, 7	syl 17	. . . . . 6 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = 𝐴)
9		f1of 6618	. . . . . . . . 9 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓:{∅}⟶𝐴)
10		0ex 5214	. . . . . . . . . . 11 ⊢ ∅ ∈ V
11	10	fsn2 6901	. . . . . . . . . 10 ⊢ (◡𝑓:{∅}⟶𝐴 ↔ ((◡𝑓‘∅) ∈ 𝐴 ∧ ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩}))
12	11	simprbi 499	. . . . . . . . 9 ⊢ (◡𝑓:{∅}⟶𝐴 → ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩})
13	9, 12	syl 17	. . . . . . . 8 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ◡𝑓 = {⟨∅, (◡𝑓‘∅)⟩})
14	13	rneqd 5811	. . . . . . 7 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = ran {⟨∅, (◡𝑓‘∅)⟩})
15	10	rnsnop 6084	. . . . . . 7 ⊢ ran {⟨∅, (◡𝑓‘∅)⟩} = {(◡𝑓‘∅)}
16	14, 15	syl6eq 2875	. . . . . 6 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → ran ◡𝑓 = {(◡𝑓‘∅)})
17	8, 16	eqtr3d 2861	. . . . 5 ⊢ (◡𝑓:{∅}–1-1-onto→𝐴 → 𝐴 = {(◡𝑓‘∅)})
18		fvex 6686	. . . . . 6 ⊢ (◡𝑓‘∅) ∈ V
19		sneq 4580	. . . . . . 7 ⊢ (𝑥 = (◡𝑓‘∅) → {𝑥} = {(◡𝑓‘∅)})
20	19	eqeq2d 2835	. . . . . 6 ⊢ (𝑥 = (◡𝑓‘∅) → (𝐴 = {𝑥} ↔ 𝐴 = {(◡𝑓‘∅)}))
21	18, 20	spcev 3610	. . . . 5 ⊢ (𝐴 = {(◡𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥})
22	5, 17, 21	3syl 18	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
23	22	exlimiv 1930	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
24	4, 23	sylbi 219	. 2 ⊢ (𝐴 ≈ 1o → ∃𝑥 𝐴 = {𝑥})
25		vex 3500	. . . . 5 ⊢ 𝑥 ∈ V
26	25	ensn1 8576	. . . 4 ⊢ {𝑥} ≈ 1o
27		breq1 5072	. . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o))
28	26, 27	mpbiri 260	. . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ≈ 1o)
29	28	exlimiv 1930	. 2 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1o)
30	24, 29	impbii 211	1 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})

Syntax hints:   ↔ wb 208   = wceq 1536  ∃wex 1779   ∈ wcel 2113  ∅c0 4294  {csn 4570  ⟨cop 4576   class class class wbr 5069  ^◡ccnv 5557  ran crn 5559  ⟶wf 6354  –onto→wfo 6356  –1-1-onto→wf1o 6357  ‘cfv 6358  1_oc1o 8098   ≈ cen 8509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-1o 8105  df-en 8513

This theorem is referenced by:  en1b  8580  reuen1  8581  en2  8757  card1  9400  pm54.43  9432  hash1elsn  13735  hash1snb  13783  ufildom1  22537  unidifsnel  30298  unidifsnne  30299  funen1cnv  32361  lfuhgr3  32370  snen1g  39896