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Theorem en1 6215
Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.)
Assertion
Ref Expression
en1 (A ≈ 1𝑜x A = {x})
Distinct variable group:   x,A

Proof of Theorem en1
Dummy variable f is distinct from all other variables.
StepHypRef Expression
1 df1o2 5952 . . . . 5 1𝑜 = {∅}
21breq2i 3763 . . . 4 (A ≈ 1𝑜A ≈ {∅})
3 bren 6164 . . . 4 (A ≈ {∅} ↔ f f:A1-1-onto→{∅})
42, 3bitri 173 . . 3 (A ≈ 1𝑜f f:A1-1-onto→{∅})
5 f1ocnv 5082 . . . . 5 (f:A1-1-onto→{∅} → f:{∅}–1-1-ontoA)
6 f1ofo 5076 . . . . . . . 8 (f:{∅}–1-1-ontoAf:{∅}–ontoA)
7 forn 5052 . . . . . . . 8 (f:{∅}–ontoA → ran f = A)
86, 7syl 14 . . . . . . 7 (f:{∅}–1-1-ontoA → ran f = A)
9 f1of 5069 . . . . . . . . . 10 (f:{∅}–1-1-ontoAf:{∅}⟶A)
10 0ex 3875 . . . . . . . . . . . 12 V
1110fsn2 5280 . . . . . . . . . . 11 (f:{∅}⟶A ↔ ((f‘∅) A f = {⟨∅, (f‘∅)⟩}))
1211simprbi 260 . . . . . . . . . 10 (f:{∅}⟶Af = {⟨∅, (f‘∅)⟩})
139, 12syl 14 . . . . . . . . 9 (f:{∅}–1-1-ontoAf = {⟨∅, (f‘∅)⟩})
1413rneqd 4506 . . . . . . . 8 (f:{∅}–1-1-ontoA → ran f = ran {⟨∅, (f‘∅)⟩})
1510rnsnop 4744 . . . . . . . 8 ran {⟨∅, (f‘∅)⟩} = {(f‘∅)}
1614, 15syl6eq 2085 . . . . . . 7 (f:{∅}–1-1-ontoA → ran f = {(f‘∅)})
178, 16eqtr3d 2071 . . . . . 6 (f:{∅}–1-1-ontoAA = {(f‘∅)})
185, 17syl 14 . . . . 5 (f:A1-1-onto→{∅} → A = {(f‘∅)})
19 f1ofn 5070 . . . . . . 7 (f:{∅}–1-1-ontoAf Fn {∅})
2010snid 3394 . . . . . . 7 {∅}
21 funfvex 5135 . . . . . . . 8 ((Fun f dom f) → (f‘∅) V)
2221funfni 4942 . . . . . . 7 ((f Fn {∅} {∅}) → (f‘∅) V)
2319, 20, 22sylancl 392 . . . . . 6 (f:{∅}–1-1-ontoA → (f‘∅) V)
24 sneq 3378 . . . . . . . 8 (x = (f‘∅) → {x} = {(f‘∅)})
2524eqeq2d 2048 . . . . . . 7 (x = (f‘∅) → (A = {x} ↔ A = {(f‘∅)}))
2625spcegv 2635 . . . . . 6 ((f‘∅) V → (A = {(f‘∅)} → x A = {x}))
2723, 26syl 14 . . . . 5 (f:{∅}–1-1-ontoA → (A = {(f‘∅)} → x A = {x}))
285, 18, 27sylc 56 . . . 4 (f:A1-1-onto→{∅} → x A = {x})
2928exlimiv 1486 . . 3 (f f:A1-1-onto→{∅} → x A = {x})
304, 29sylbi 114 . 2 (A ≈ 1𝑜x A = {x})
31 vex 2554 . . . . 5 x V
3231ensn1 6212 . . . 4 {x} ≈ 1𝑜
33 breq1 3758 . . . 4 (A = {x} → (A ≈ 1𝑜 ↔ {x} ≈ 1𝑜))
3432, 33mpbiri 157 . . 3 (A = {x} → A ≈ 1𝑜)
3534exlimiv 1486 . 2 (x A = {x} → A ≈ 1𝑜)
3630, 35impbii 117 1 (A ≈ 1𝑜x A = {x})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  c0 3218  {csn 3367  cop 3370   class class class wbr 3755  ccnv 4287  ran crn 4289   Fn wfn 4840  wf 4841  ontowfo 4843  1-1-ontowf1o 4844  cfv 4845  1𝑜c1o 5933  cen 6155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-1o 5940  df-en 6158
This theorem is referenced by:  en1bg  6216  reuen1  6217
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