Theorem en0OLD 9075

Description: Obsolete version of en0 9074 as of 23-Sep-2024. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5386. (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.

Step	Hyp	Ref	Expression
1		bren 9009	. . 3 ⊢ (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴–1-1-onto→∅)
2		f1ocnv 6873	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→∅ → ◡𝑓:∅–1-1-onto→𝐴)
3		f1o00 6896	. . . . . 6 ⊢ (◡𝑓:∅–1-1-onto→𝐴 ↔ (◡𝑓 = ∅ ∧ 𝐴 = ∅))
4	3	simprbi 496	. . . . 5 ⊢ (◡𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅)
5	2, 4	syl 17	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅)
6	5	exlimiv 1929	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅)
7	1, 6	sylbi 217	. 2 ⊢ (𝐴 ≈ ∅ → 𝐴 = ∅)
8		0ex 5328	. . . . 5 ⊢ ∅ ∈ V
9		f1oeq1 6849	. . . . 5 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
10		f1o0 6898	. . . . 5 ⊢ ∅:∅–1-1-onto→∅
11	8, 9, 10	ceqsexv2d 3540	. . . 4 ⊢ ∃𝑓 𝑓:∅–1-1-onto→∅
12		bren 9009	. . . 4 ⊢ (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅)
13	11, 12	mpbir 231	. . 3 ⊢ ∅ ≈ ∅
14		breq1 5172	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ≈ ∅ ↔ ∅ ≈ ∅))
15	13, 14	mpbiri 258	. 2 ⊢ (𝐴 = ∅ → 𝐴 ≈ ∅)
16	7, 15	impbii 209	1 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 206   = wceq 1537  ∃wex 1777  ∅c0 4347   class class class wbr 5169  ^◡ccnv 5698  –1-1-onto→wf1o 6571   ≈ cen 8996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450  ax-un 7766

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-mo 2537  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-en 9000

This theorem is referenced by: (None)