Theorem en0ALT 9074

Description: Shorter proof of en0 9073, depending on ax-pow 5331 and ax-un 7775. (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.

Step	Hyp	Ref	Expression
1		bren 9011	. . 3 ⊢ (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴⟶1-1onto∅)
2		f1ocnv 6859	. . . . 5 ⊢ (𝑓:𝐴⟶1-1onto∅ → ◡𝑓:∅⟶1-1onto𝐴)
3		f1o00 6883	. . . . . 6 ⊢ (◡𝑓:∅⟶1-1onto𝐴 ↔ (◡𝑓 = ∅ ∧ 𝐴 = ∅))
4	3	simprbi 503	. . . . 5 ⊢ (◡𝑓:∅⟶1-1onto𝐴 → 𝐴 = ∅)
5	2, 4	syl 18	. . . 4 ⊢ (𝑓:𝐴⟶1-1onto∅ → 𝐴 = ∅)
6	5	exlimiv 1965	. . 3 ⊢ (∃𝑓 𝑓:𝐴⟶1-1onto∅ → 𝐴 = ∅)
7	1, 6	sylbi 220	. 2 ⊢ (𝐴 ≈ ∅ → 𝐴 = ∅)
8		0ex 5264	. . . 4 ⊢ ∅ ∈ V
9	8	enref 9040	. . 3 ⊢ ∅ ≈ ∅
10		breq1 5109	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ≈ ∅ ↔ ∅ ≈ ∅))
11	9, 10	mpbiri 261	. 2 ⊢ (𝐴 = ∅ → 𝐴 ≈ ∅)
12	7, 11	impbii 212	1 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  ∃wex 1812  ∅c0 4281   class class class wbr 5106  ^◡ccnv 5654  ⟶1-1ontowf1o 6557   ≈ cen 8998

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-en 9002

This theorem is used by: (None)