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Theorem en0r 9083
Description: The empty set is equinumerous only to itself. (Contributed by BTernaryTau, 29-Nov-2024.)
Assertion
Ref Expression
en0r (∅ ≈ 𝐴𝐴 = ∅)

Proof of Theorem en0r
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 encv 9013 . . . . 5 (∅ ≈ 𝐴 → (∅ ∈ V ∧ 𝐴 ∈ V))
2 breng 9014 . . . . 5 ((∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ≈ 𝐴 ↔ ∃𝑓 𝑓:∅–1-1-onto𝐴))
31, 2syl 17 . . . 4 (∅ ≈ 𝐴 → (∅ ≈ 𝐴 ↔ ∃𝑓 𝑓:∅–1-1-onto𝐴))
43ibi 267 . . 3 (∅ ≈ 𝐴 → ∃𝑓 𝑓:∅–1-1-onto𝐴)
5 f1o00 6899 . . . . 5 (𝑓:∅–1-1-onto𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
65simprbi 496 . . . 4 (𝑓:∅–1-1-onto𝐴𝐴 = ∅)
76exlimiv 1929 . . 3 (∃𝑓 𝑓:∅–1-1-onto𝐴𝐴 = ∅)
84, 7syl 17 . 2 (∅ ≈ 𝐴𝐴 = ∅)
9 0ex 5325 . . . . 5 ∅ ∈ V
10 f1oeq1 6852 . . . . 5 (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
11 f1o0 6901 . . . . 5 ∅:∅–1-1-onto→∅
129, 10, 11ceqsexv2d 3545 . . . 4 𝑓 𝑓:∅–1-1-onto→∅
13 breng 9014 . . . . 5 ((∅ ∈ V ∧ ∅ ∈ V) → (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅))
149, 9, 13mp2an 691 . . . 4 (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅)
1512, 14mpbir 231 . . 3 ∅ ≈ ∅
16 breq2 5170 . . 3 (𝐴 = ∅ → (∅ ≈ 𝐴 ↔ ∅ ≈ ∅))
1715, 16mpbiri 258 . 2 (𝐴 = ∅ → ∅ ≈ 𝐴)
188, 17impbii 209 1 (∅ ≈ 𝐴𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wex 1777  wcel 2108  Vcvv 3488  c0 4352   class class class wbr 5166  1-1-ontowf1o 6574  cen 9002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-en 9006
This theorem is referenced by:  0sdomg  9172  fiint  9396  rp-isfinite6  43482  ensucne0  43493
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