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Theorem en1 7974
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 7524 . . . . 5 1𝑜 = {∅}
21breq2i 4626 . . . 4 (𝐴 ≈ 1𝑜𝐴 ≈ {∅})
3 bren 7915 . . . 4 (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴1-1-onto→{∅})
42, 3bitri 264 . . 3 (𝐴 ≈ 1𝑜 ↔ ∃𝑓 𝑓:𝐴1-1-onto→{∅})
5 f1ocnv 6111 . . . . 5 (𝑓:𝐴1-1-onto→{∅} → 𝑓:{∅}–1-1-onto𝐴)
6 f1ofo 6106 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴𝑓:{∅}–onto𝐴)
7 forn 6080 . . . . . . 7 (𝑓:{∅}–onto𝐴 → ran 𝑓 = 𝐴)
86, 7syl 17 . . . . . 6 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = 𝐴)
9 f1of 6099 . . . . . . . . 9 (𝑓:{∅}–1-1-onto𝐴𝑓:{∅}⟶𝐴)
10 0ex 4755 . . . . . . . . . . 11 ∅ ∈ V
1110fsn2 6363 . . . . . . . . . 10 (𝑓:{∅}⟶𝐴 ↔ ((𝑓‘∅) ∈ 𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩}))
1211simprbi 480 . . . . . . . . 9 (𝑓:{∅}⟶𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩})
139, 12syl 17 . . . . . . . 8 (𝑓:{∅}–1-1-onto𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩})
1413rneqd 5318 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = ran {⟨∅, (𝑓‘∅)⟩})
1510rnsnop 5580 . . . . . . 7 ran {⟨∅, (𝑓‘∅)⟩} = {(𝑓‘∅)}
1614, 15syl6eq 2671 . . . . . 6 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = {(𝑓‘∅)})
178, 16eqtr3d 2657 . . . . 5 (𝑓:{∅}–1-1-onto𝐴𝐴 = {(𝑓‘∅)})
18 fvex 6163 . . . . . 6 (𝑓‘∅) ∈ V
19 sneq 4163 . . . . . . 7 (𝑥 = (𝑓‘∅) → {𝑥} = {(𝑓‘∅)})
2019eqeq2d 2631 . . . . . 6 (𝑥 = (𝑓‘∅) → (𝐴 = {𝑥} ↔ 𝐴 = {(𝑓‘∅)}))
2118, 20spcev 3289 . . . . 5 (𝐴 = {(𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥})
225, 17, 213syl 18 . . . 4 (𝑓:𝐴1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
2322exlimiv 1855 . . 3 (∃𝑓 𝑓:𝐴1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
244, 23sylbi 207 . 2 (𝐴 ≈ 1𝑜 → ∃𝑥 𝐴 = {𝑥})
25 vex 3192 . . . . 5 𝑥 ∈ V
2625ensn1 7971 . . . 4 {𝑥} ≈ 1𝑜
27 breq1 4621 . . . 4 (𝐴 = {𝑥} → (𝐴 ≈ 1𝑜 ↔ {𝑥} ≈ 1𝑜))
2826, 27mpbiri 248 . . 3 (𝐴 = {𝑥} → 𝐴 ≈ 1𝑜)
2928exlimiv 1855 . 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 1480  wex 1701  wcel 1987  c0 3896  {csn 4153  cop 4159   class class class wbr 4618  ccnv 5078  ran crn 5080  wf 5848  ontowfo 5850  1-1-ontowf1o 5851  cfv 5852  1𝑜c1o 7505  cen 7903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-1o 7512  df-en 7907
This theorem is referenced by:  en1b  7975  reuen1  7976  en2  8147  card1  8745  pm54.43  8777  hash1snb  13154  ufildom1  21649
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