Theorem en0ALT 9024

Description: Shorter proof of en0 9023, depending on ax-pow 5326 and ax-un 7734. (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 8961	. . 3 ⊢ (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴–1-1-onto→∅)
2		f1ocnv 6825	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→∅ → ◡𝑓:∅–1-1-onto→𝐴)
3		f1o00 6848	. . . . . 6 ⊢ (◡𝑓:∅–1-1-onto→𝐴 ↔ (◡𝑓 = ∅ ∧ 𝐴 = ∅))
4	3	simprbi 503	. . . . 5 ⊢ (◡𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅)
5	2, 4	syl 18	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅)
6	5	exlimiv 1963	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅)
7	1, 6	sylbi 220	. 2 ⊢ (𝐴 ≈ ∅ → 𝐴 = ∅)
8		0ex 5260	. . . 4 ⊢ ∅ ∈ V
9	8	enref 8990	. . 3 ⊢ ∅ ≈ ∅
10		breq1 5105	. . 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 4278   class class class wbr 5102  ^◡ccnv 5646  –1-1-onto→wf1o 6526   ≈ cen 8948

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

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 2100  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-en 8952

This theorem is used by: (None)