ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  en1 GIF version

Theorem en1 6516
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 6194 . . . . 5 1𝑜 = {∅}
21breq2i 3853 . . . 4 (𝐴 ≈ 1𝑜𝐴 ≈ {∅})
3 bren 6464 . . . 4 (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴1-1-onto→{∅})
42, 3bitri 182 . . 3 (𝐴 ≈ 1𝑜 ↔ ∃𝑓 𝑓:𝐴1-1-onto→{∅})
5 f1ocnv 5266 . . . . 5 (𝑓:𝐴1-1-onto→{∅} → 𝑓:{∅}–1-1-onto𝐴)
6 f1ofo 5260 . . . . . . . 8 (𝑓:{∅}–1-1-onto𝐴𝑓:{∅}–onto𝐴)
7 forn 5236 . . . . . . . 8 (𝑓:{∅}–onto𝐴 → ran 𝑓 = 𝐴)
86, 7syl 14 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = 𝐴)
9 f1of 5253 . . . . . . . . . 10 (𝑓:{∅}–1-1-onto𝐴𝑓:{∅}⟶𝐴)
10 0ex 3966 . . . . . . . . . . . 12 ∅ ∈ V
1110fsn2 5471 . . . . . . . . . . 11 (𝑓:{∅}⟶𝐴 ↔ ((𝑓‘∅) ∈ 𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩}))
1211simprbi 269 . . . . . . . . . 10 (𝑓:{∅}⟶𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩})
139, 12syl 14 . . . . . . . . 9 (𝑓:{∅}–1-1-onto𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩})
1413rneqd 4664 . . . . . . . 8 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = ran {⟨∅, (𝑓‘∅)⟩})
1510rnsnop 4911 . . . . . . . 8 ran {⟨∅, (𝑓‘∅)⟩} = {(𝑓‘∅)}
1614, 15syl6eq 2136 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = {(𝑓‘∅)})
178, 16eqtr3d 2122 . . . . . 6 (𝑓:{∅}–1-1-onto𝐴𝐴 = {(𝑓‘∅)})
185, 17syl 14 . . . . 5 (𝑓:𝐴1-1-onto→{∅} → 𝐴 = {(𝑓‘∅)})
19 f1ofn 5254 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴𝑓 Fn {∅})
2010snid 3475 . . . . . . 7 ∅ ∈ {∅}
21 funfvex 5322 . . . . . . . 8 ((Fun 𝑓 ∧ ∅ ∈ dom 𝑓) → (𝑓‘∅) ∈ V)
2221funfni 5114 . . . . . . 7 ((𝑓 Fn {∅} ∧ ∅ ∈ {∅}) → (𝑓‘∅) ∈ V)
2319, 20, 22sylancl 404 . . . . . 6 (𝑓:{∅}–1-1-onto𝐴 → (𝑓‘∅) ∈ V)
24 sneq 3457 . . . . . . . 8 (𝑥 = (𝑓‘∅) → {𝑥} = {(𝑓‘∅)})
2524eqeq2d 2099 . . . . . . 7 (𝑥 = (𝑓‘∅) → (𝐴 = {𝑥} ↔ 𝐴 = {(𝑓‘∅)}))
2625spcegv 2707 . . . . . 6 ((𝑓‘∅) ∈ V → (𝐴 = {(𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥}))
2723, 26syl 14 . . . . 5 (𝑓:{∅}–1-1-onto𝐴 → (𝐴 = {(𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥}))
285, 18, 27sylc 61 . . . 4 (𝑓:𝐴1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
2928exlimiv 1534 . . 3 (∃𝑓 𝑓:𝐴1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
304, 29sylbi 119 . 2 (𝐴 ≈ 1𝑜 → ∃𝑥 𝐴 = {𝑥})
31 vex 2622 . . . . 5 𝑥 ∈ V
3231ensn1 6513 . . . 4 {𝑥} ≈ 1𝑜
33 breq1 3848 . . . 4 (𝐴 = {𝑥} → (𝐴 ≈ 1𝑜 ↔ {𝑥} ≈ 1𝑜))
3432, 33mpbiri 166 . . 3 (𝐴 = {𝑥} → 𝐴 ≈ 1𝑜)
3534exlimiv 1534 . 2 (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1𝑜)
3630, 35impbii 124 1 (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  wex 1426  wcel 1438  Vcvv 2619  c0 3286  {csn 3446  cop 3449   class class class wbr 3845  ccnv 4437  ran crn 4439   Fn wfn 5010  wf 5011  ontowfo 5013  1-1-ontowf1o 5014  cfv 5015  1𝑜c1o 6174  cen 6455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-suc 4198  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-1o 6181  df-en 6458
This theorem is referenced by:  en1bg  6517  reuen1  6518  pm54.43  6818
  Copyright terms: Public domain W3C validator