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Theorem en0OLD 8621
Description: Obsolete version of en0 8620 as of 31-Jul-2024. (Contributed by NM, 27-May-1998.) (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 8566 . . 3 (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴1-1-onto→∅)
2 f1ocnv 6632 . . . . 5 (𝑓:𝐴1-1-onto→∅ → 𝑓:∅–1-1-onto𝐴)
3 f1o00 6654 . . . . . 6 (𝑓:∅–1-1-onto𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
43simprbi 500 . . . . 5 (𝑓:∅–1-1-onto𝐴𝐴 = ∅)
52, 4syl 17 . . . 4 (𝑓:𝐴1-1-onto→∅ → 𝐴 = ∅)
65exlimiv 1937 . . 3 (∃𝑓 𝑓:𝐴1-1-onto→∅ → 𝐴 = ∅)
71, 6sylbi 220 . 2 (𝐴 ≈ ∅ → 𝐴 = ∅)
8 0ex 5175 . . . 4 ∅ ∈ V
98enref 8590 . . 3 ∅ ≈ ∅
10 breq1 5033 . . 3 (𝐴 = ∅ → (𝐴 ≈ ∅ ↔ ∅ ≈ ∅))
119, 10mpbiri 261 . 2 (𝐴 = ∅ → 𝐴 ≈ ∅)
127, 11impbii 212 1 (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1542  wex 1786  c0 4211   class class class wbr 5030  ccnv 5524  1-1-ontowf1o 6338  cen 8554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-en 8558
This theorem is referenced by: (None)
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