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Theorem en0OLD 9030
Description: Obsolete version of en0 9029 as of 23-Sep-2024. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5359. (Revised by BTernaryTau, 31-Jul-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
en0OLD (𝐴 ≈ ∅ ↔ 𝐴 = ∅)

Proof of Theorem en0OLD
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 bren 8965 . . 3 (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴1-1-onto→∅)
2 f1ocnv 6845 . . . . 5 (𝑓:𝐴1-1-onto→∅ → 𝑓:∅–1-1-onto𝐴)
3 f1o00 6868 . . . . . 6 (𝑓:∅–1-1-onto𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
43simprbi 496 . . . . 5 (𝑓:∅–1-1-onto𝐴𝐴 = ∅)
52, 4syl 17 . . . 4 (𝑓:𝐴1-1-onto→∅ → 𝐴 = ∅)
65exlimiv 1926 . . 3 (∃𝑓 𝑓:𝐴1-1-onto→∅ → 𝐴 = ∅)
71, 6sylbi 216 . 2 (𝐴 ≈ ∅ → 𝐴 = ∅)
8 0ex 5301 . . . . 5 ∅ ∈ V
9 f1oeq1 6821 . . . . 5 (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
10 f1o0 6870 . . . . 5 ∅:∅–1-1-onto→∅
118, 9, 10ceqsexv2d 3524 . . . 4 𝑓 𝑓:∅–1-1-onto→∅
12 bren 8965 . . . 4 (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅)
1311, 12mpbir 230 . . 3 ∅ ≈ ∅
14 breq1 5145 . . 3 (𝐴 = ∅ → (𝐴 ≈ ∅ ↔ ∅ ≈ ∅))
1513, 14mpbiri 258 . 2 (𝐴 = ∅ → 𝐴 ≈ ∅)
167, 15impbii 208 1 (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1534  wex 1774  c0 4318   class class class wbr 5142  ccnv 5671  1-1-ontowf1o 6541  cen 8952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-mo 2529  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-en 8956
This theorem is referenced by: (None)
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