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Theorem en1 7909
 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.
StepHypRef Expression
1 df1o2 7459 . . . . 5 1𝑜 = {∅}
21breq2i 4591 . . . 4 (𝐴 ≈ 1𝑜𝐴 ≈ {∅})
3 bren 7850 . . . 4 (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴1-1-onto→{∅})
42, 3bitri 263 . . 3 (𝐴 ≈ 1𝑜 ↔ ∃𝑓 𝑓:𝐴1-1-onto→{∅})
5 f1ocnv 6062 . . . . 5 (𝑓:𝐴1-1-onto→{∅} → 𝑓:{∅}–1-1-onto𝐴)
6 f1ofo 6057 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴𝑓:{∅}–onto𝐴)
7 forn 6031 . . . . . . 7 (𝑓:{∅}–onto𝐴 → ran 𝑓 = 𝐴)
86, 7syl 17 . . . . . 6 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = 𝐴)
9 f1of 6050 . . . . . . . . 9 (𝑓:{∅}–1-1-onto𝐴𝑓:{∅}⟶𝐴)
10 0ex 4718 . . . . . . . . . . 11 ∅ ∈ V
1110fsn2 6309 . . . . . . . . . 10 (𝑓:{∅}⟶𝐴 ↔ ((𝑓‘∅) ∈ 𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩}))
1211simprbi 479 . . . . . . . . 9 (𝑓:{∅}⟶𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩})
139, 12syl 17 . . . . . . . 8 (𝑓:{∅}–1-1-onto𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩})
1413rneqd 5274 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = ran {⟨∅, (𝑓‘∅)⟩})
1510rnsnop 5534 . . . . . . 7 ran {⟨∅, (𝑓‘∅)⟩} = {(𝑓‘∅)}
1614, 15syl6eq 2660 . . . . . 6 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = {(𝑓‘∅)})
178, 16eqtr3d 2646 . . . . 5 (𝑓:{∅}–1-1-onto𝐴𝐴 = {(𝑓‘∅)})
18 fvex 6113 . . . . . 6 (𝑓‘∅) ∈ V
19 sneq 4135 . . . . . . 7 (𝑥 = (𝑓‘∅) → {𝑥} = {(𝑓‘∅)})
2019eqeq2d 2620 . . . . . 6 (𝑥 = (𝑓‘∅) → (𝐴 = {𝑥} ↔ 𝐴 = {(𝑓‘∅)}))
2118, 20spcev 3273 . . . . 5 (𝐴 = {(𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥})
225, 17, 213syl 18 . . . 4 (𝑓:𝐴1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
2322exlimiv 1845 . . 3 (∃𝑓 𝑓:𝐴1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
244, 23sylbi 206 . 2 (𝐴 ≈ 1𝑜 → ∃𝑥 𝐴 = {𝑥})
25 vex 3176 . . . . 5 𝑥 ∈ V
2625ensn1 7906 . . . 4 {𝑥} ≈ 1𝑜
27 breq1 4586 . . . 4 (𝐴 = {𝑥} → (𝐴 ≈ 1𝑜 ↔ {𝑥} ≈ 1𝑜))
2826, 27mpbiri 247 . . 3 (𝐴 = {𝑥} → 𝐴 ≈ 1𝑜)
2928exlimiv 1845 . 2 (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1𝑜)
3024, 29impbii 198 1 (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∅c0 3874  {csn 4125  ⟨cop 4131   class class class wbr 4583  ◡ccnv 5037  ran crn 5039  ⟶wf 5800  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  1𝑜c1o 7440   ≈ cen 7838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1o 7447  df-en 7842 This theorem is referenced by:  en1b  7910  reuen1  7911  en2  8081  card1  8677  pm54.43  8709  hash1snb  13068  ufildom1  21540
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