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

Theorem en1 6799
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 6430 . . . . 5 1o = {∅}
21breq2i 4012 . . . 4 (𝐴 ≈ 1o𝐴 ≈ {∅})
3 bren 6747 . . . 4 (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴1-1-onto→{∅})
42, 3bitri 184 . . 3 (𝐴 ≈ 1o ↔ ∃𝑓 𝑓:𝐴1-1-onto→{∅})
5 f1ocnv 5475 . . . . 5 (𝑓:𝐴1-1-onto→{∅} → 𝑓:{∅}–1-1-onto𝐴)
6 f1ofo 5469 . . . . . . . 8 (𝑓:{∅}–1-1-onto𝐴𝑓:{∅}–onto𝐴)
7 forn 5442 . . . . . . . 8 (𝑓:{∅}–onto𝐴 → ran 𝑓 = 𝐴)
86, 7syl 14 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = 𝐴)
9 f1of 5462 . . . . . . . . . 10 (𝑓:{∅}–1-1-onto𝐴𝑓:{∅}⟶𝐴)
10 0ex 4131 . . . . . . . . . . . 12 ∅ ∈ V
1110fsn2 5691 . . . . . . . . . . 11 (𝑓:{∅}⟶𝐴 ↔ ((𝑓‘∅) ∈ 𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩}))
1211simprbi 275 . . . . . . . . . 10 (𝑓:{∅}⟶𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩})
139, 12syl 14 . . . . . . . . 9 (𝑓:{∅}–1-1-onto𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩})
1413rneqd 4857 . . . . . . . 8 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = ran {⟨∅, (𝑓‘∅)⟩})
1510rnsnop 5110 . . . . . . . 8 ran {⟨∅, (𝑓‘∅)⟩} = {(𝑓‘∅)}
1614, 15eqtrdi 2226 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = {(𝑓‘∅)})
178, 16eqtr3d 2212 . . . . . 6 (𝑓:{∅}–1-1-onto𝐴𝐴 = {(𝑓‘∅)})
185, 17syl 14 . . . . 5 (𝑓:𝐴1-1-onto→{∅} → 𝐴 = {(𝑓‘∅)})
19 f1ofn 5463 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴𝑓 Fn {∅})
2010snid 3624 . . . . . . 7 ∅ ∈ {∅}
21 funfvex 5533 . . . . . . . 8 ((Fun 𝑓 ∧ ∅ ∈ dom 𝑓) → (𝑓‘∅) ∈ V)
2221funfni 5317 . . . . . . 7 ((𝑓 Fn {∅} ∧ ∅ ∈ {∅}) → (𝑓‘∅) ∈ V)
2319, 20, 22sylancl 413 . . . . . 6 (𝑓:{∅}–1-1-onto𝐴 → (𝑓‘∅) ∈ V)
24 sneq 3604 . . . . . . . 8 (𝑥 = (𝑓‘∅) → {𝑥} = {(𝑓‘∅)})
2524eqeq2d 2189 . . . . . . 7 (𝑥 = (𝑓‘∅) → (𝐴 = {𝑥} ↔ 𝐴 = {(𝑓‘∅)}))
2625spcegv 2826 . . . . . 6 ((𝑓‘∅) ∈ V → (𝐴 = {(𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥}))
2723, 26syl 14 . . . . 5 (𝑓:{∅}–1-1-onto𝐴 → (𝐴 = {(𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥}))
285, 18, 27sylc 62 . . . 4 (𝑓:𝐴1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
2928exlimiv 1598 . . 3 (∃𝑓 𝑓:𝐴1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
304, 29sylbi 121 . 2 (𝐴 ≈ 1o → ∃𝑥 𝐴 = {𝑥})
31 vex 2741 . . . . 5 𝑥 ∈ V
3231ensn1 6796 . . . 4 {𝑥} ≈ 1o
33 breq1 4007 . . . 4 (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o))
3432, 33mpbiri 168 . . 3 (𝐴 = {𝑥} → 𝐴 ≈ 1o)
3534exlimiv 1598 . 2 (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1o)
3630, 35impbii 126 1 (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wex 1492  wcel 2148  Vcvv 2738  c0 3423  {csn 3593  cop 3596   class class class wbr 4004  ccnv 4626  ran crn 4628   Fn wfn 5212  wf 5213  ontowfo 5215  1-1-ontowf1o 5216  cfv 5217  1oc1o 6410  cen 6738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-suc 4372  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-1o 6417  df-en 6741
This theorem is referenced by:  en1bg  6800  reuen1  6801  pm54.43  7189
  Copyright terms: Public domain W3C validator