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Theorem en0OLD 9043
Description: Obsolete version of en0 9042 as of 23-Sep-2024. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5367. (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 8978 . . 3 (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴1-1-onto→∅)
2 f1ocnv 6854 . . . . 5 (𝑓:𝐴1-1-onto→∅ → 𝑓:∅–1-1-onto𝐴)
3 f1o00 6877 . . . . . 6 (𝑓:∅–1-1-onto𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
43simprbi 495 . . . . 5 (𝑓:∅–1-1-onto𝐴𝐴 = ∅)
52, 4syl 17 . . . 4 (𝑓:𝐴1-1-onto→∅ → 𝐴 = ∅)
65exlimiv 1925 . . 3 (∃𝑓 𝑓:𝐴1-1-onto→∅ → 𝐴 = ∅)
71, 6sylbi 216 . 2 (𝐴 ≈ ∅ → 𝐴 = ∅)
8 0ex 5309 . . . . 5 ∅ ∈ V
9 f1oeq1 6830 . . . . 5 (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
10 f1o0 6879 . . . . 5 ∅:∅–1-1-onto→∅
118, 9, 10ceqsexv2d 3526 . . . 4 𝑓 𝑓:∅–1-1-onto→∅
12 bren 8978 . . . 4 (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅)
1311, 12mpbir 230 . . 3 ∅ ≈ ∅
14 breq1 5153 . . 3 (𝐴 = ∅ → (𝐴 ≈ ∅ ↔ ∅ ≈ ∅))
1513, 14mpbiri 257 . 2 (𝐴 = ∅ → 𝐴 ≈ ∅)
167, 15impbii 208 1 (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wex 1773  c0 4324   class class class wbr 5150  ccnv 5679  1-1-ontowf1o 6550  cen 8965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2529  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-en 8969
This theorem is referenced by: (None)
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