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Theorem en0ALT 8962
Description: Shorter proof of en0 8960, depending on ax-pow 5321 and ax-un 7673. (Contributed by NM, 27-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
en0ALT (𝐴 ≈ ∅ ↔ 𝐴 = ∅)

Proof of Theorem en0ALT
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 bren 8896 . . 3 (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴1-1-onto→∅)
2 f1ocnv 6797 . . . . 5 (𝑓:𝐴1-1-onto→∅ → 𝑓:∅–1-1-onto𝐴)
3 f1o00 6820 . . . . . 6 (𝑓:∅–1-1-onto𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
43simprbi 498 . . . . 5 (𝑓:∅–1-1-onto𝐴𝐴 = ∅)
52, 4syl 17 . . . 4 (𝑓:𝐴1-1-onto→∅ → 𝐴 = ∅)
65exlimiv 1934 . . 3 (∃𝑓 𝑓:𝐴1-1-onto→∅ → 𝐴 = ∅)
71, 6sylbi 216 . 2 (𝐴 ≈ ∅ → 𝐴 = ∅)
8 0ex 5265 . . . 4 ∅ ∈ V
98enref 8928 . . 3 ∅ ≈ ∅
10 breq1 5109 . . 3 (𝐴 = ∅ → (𝐴 ≈ ∅ ↔ ∅ ≈ ∅))
119, 10mpbiri 258 . 2 (𝐴 = ∅ → 𝐴 ≈ ∅)
127, 11impbii 208 1 (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wex 1782  c0 4283   class class class wbr 5106  ccnv 5633  1-1-ontowf1o 6496  cen 8883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-en 8887
This theorem is referenced by: (None)
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