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Theorem en1 8175
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 7725 . . . . 5 1𝑜 = {∅}
21breq2i 4792 . . . 4 (𝐴 ≈ 1𝑜𝐴 ≈ {∅})
3 bren 8117 . . . 4 (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴1-1-onto→{∅})
42, 3bitri 264 . . 3 (𝐴 ≈ 1𝑜 ↔ ∃𝑓 𝑓:𝐴1-1-onto→{∅})
5 f1ocnv 6290 . . . . 5 (𝑓:𝐴1-1-onto→{∅} → 𝑓:{∅}–1-1-onto𝐴)
6 f1ofo 6285 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴𝑓:{∅}–onto𝐴)
7 forn 6259 . . . . . . 7 (𝑓:{∅}–onto𝐴 → ran 𝑓 = 𝐴)
86, 7syl 17 . . . . . 6 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = 𝐴)
9 f1of 6278 . . . . . . . . 9 (𝑓:{∅}–1-1-onto𝐴𝑓:{∅}⟶𝐴)
10 0ex 4921 . . . . . . . . . . 11 ∅ ∈ V
1110fsn2 6545 . . . . . . . . . 10 (𝑓:{∅}⟶𝐴 ↔ ((𝑓‘∅) ∈ 𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩}))
1211simprbi 478 . . . . . . . . 9 (𝑓:{∅}⟶𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩})
139, 12syl 17 . . . . . . . 8 (𝑓:{∅}–1-1-onto𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩})
1413rneqd 5491 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = ran {⟨∅, (𝑓‘∅)⟩})
1510rnsnop 5759 . . . . . . 7 ran {⟨∅, (𝑓‘∅)⟩} = {(𝑓‘∅)}
1614, 15syl6eq 2820 . . . . . 6 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = {(𝑓‘∅)})
178, 16eqtr3d 2806 . . . . 5 (𝑓:{∅}–1-1-onto𝐴𝐴 = {(𝑓‘∅)})
18 fvex 6342 . . . . . 6 (𝑓‘∅) ∈ V
19 sneq 4324 . . . . . . 7 (𝑥 = (𝑓‘∅) → {𝑥} = {(𝑓‘∅)})
2019eqeq2d 2780 . . . . . 6 (𝑥 = (𝑓‘∅) → (𝐴 = {𝑥} ↔ 𝐴 = {(𝑓‘∅)}))
2118, 20spcev 3449 . . . . 5 (𝐴 = {(𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥})
225, 17, 213syl 18 . . . 4 (𝑓:𝐴1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
2322exlimiv 2009 . . 3 (∃𝑓 𝑓:𝐴1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
244, 23sylbi 207 . 2 (𝐴 ≈ 1𝑜 → ∃𝑥 𝐴 = {𝑥})
25 vex 3352 . . . . 5 𝑥 ∈ V
2625ensn1 8172 . . . 4 {𝑥} ≈ 1𝑜
27 breq1 4787 . . . 4 (𝐴 = {𝑥} → (𝐴 ≈ 1𝑜 ↔ {𝑥} ≈ 1𝑜))
2826, 27mpbiri 248 . . 3 (𝐴 = {𝑥} → 𝐴 ≈ 1𝑜)
2928exlimiv 2009 . 2 (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1𝑜)
3024, 29impbii 199 1 (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1630  wex 1851  wcel 2144  c0 4061  {csn 4314  cop 4320   class class class wbr 4784  ccnv 5248  ran crn 5250  wf 6027  ontowfo 6029  1-1-ontowf1o 6030  cfv 6031  1𝑜c1o 7705  cen 8105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-1o 7712  df-en 8109
This theorem is referenced by:  en1b  8176  reuen1  8177  en2  8351  card1  8993  pm54.43  9025  hash1snb  13408  ufildom1  21949
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