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Theorem en0 9058
Description: The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5365, ax-un 7755. (Revised by BTernaryTau, 23-Sep-2024.)
Assertion
Ref Expression
en0 (𝐴 ≈ ∅ ↔ 𝐴 = ∅)

Proof of Theorem en0
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 encv 8993 . . . . 5 (𝐴 ≈ ∅ → (𝐴 ∈ V ∧ ∅ ∈ V))
2 breng 8994 . . . . 5 ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴1-1-onto→∅))
31, 2syl 17 . . . 4 (𝐴 ≈ ∅ → (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴1-1-onto→∅))
43ibi 267 . . 3 (𝐴 ≈ ∅ → ∃𝑓 𝑓:𝐴1-1-onto→∅)
5 f1ocnv 6860 . . . . 5 (𝑓:𝐴1-1-onto→∅ → 𝑓:∅–1-1-onto𝐴)
6 f1o00 6883 . . . . . 6 (𝑓:∅–1-1-onto𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
76simprbi 496 . . . . 5 (𝑓:∅–1-1-onto𝐴𝐴 = ∅)
85, 7syl 17 . . . 4 (𝑓:𝐴1-1-onto→∅ → 𝐴 = ∅)
98exlimiv 1930 . . 3 (∃𝑓 𝑓:𝐴1-1-onto→∅ → 𝐴 = ∅)
104, 9syl 17 . 2 (𝐴 ≈ ∅ → 𝐴 = ∅)
11 0ex 5307 . . . . 5 ∅ ∈ V
12 f1oeq1 6836 . . . . 5 (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
13 f1o0 6885 . . . . 5 ∅:∅–1-1-onto→∅
1411, 12, 13ceqsexv2d 3533 . . . 4 𝑓 𝑓:∅–1-1-onto→∅
15 breng 8994 . . . . 5 ((∅ ∈ V ∧ ∅ ∈ V) → (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅))
1611, 11, 15mp2an 692 . . . 4 (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅)
1714, 16mpbir 231 . . 3 ∅ ≈ ∅
18 breq1 5146 . . 3 (𝐴 = ∅ → (𝐴 ≈ ∅ ↔ ∅ ≈ ∅))
1917, 18mpbiri 258 . 2 (𝐴 = ∅ → 𝐴 ≈ ∅)
2010, 19impbii 209 1 (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  c0 4333   class class class wbr 5143  ccnv 5684  1-1-ontowf1o 6560  cen 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-en 8986
This theorem is referenced by:  0fi  9082  snfiOLD  9084  enrefnn  9087  dom0  9142  dom0OLD  9143  0sdomgOLD  9145  sdom0  9148  findcard  9203  findcard2  9204  nneneq  9246  nneneqOLD  9258  snnen2oOLD  9264  enp1iOLD  9314  fiintOLD  9367  cantnff  9714  cantnf0  9715  cantnfp1lem2  9719  cantnflem1  9729  cantnf  9733  cnfcom2lem  9741  cardnueq0  10004  infmap2  10257  fin23lem26  10365  cardeq0  10592  hasheq0  14402  mreexexd  17691  pmtrfmvdn0  19480  pmtrsn  19537  rp-isfinite6  43531  ensucne0OLD  43543
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