Theorem en0OLD 8670

Description: Obsolete version of en0 8669 as of 23-Sep-2024. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5243. (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 8614	. . 3 ⊢ (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴–1-1-onto→∅)
2		f1ocnv 6651	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→∅ → ◡𝑓:∅–1-1-onto→𝐴)
3		f1o00 6673	. . . . . 6 ⊢ (◡𝑓:∅–1-1-onto→𝐴 ↔ (◡𝑓 = ∅ ∧ 𝐴 = ∅))
4	3	simprbi 500	. . . . 5 ⊢ (◡𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅)
5	2, 4	syl 17	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅)
6	5	exlimiv 1938	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅)
7	1, 6	sylbi 220	. 2 ⊢ (𝐴 ≈ ∅ → 𝐴 = ∅)
8		0ex 5185	. . . . 5 ⊢ ∅ ∈ V
9		f1oeq1 6627	. . . . 5 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
10		f1o0 6675	. . . . 5 ⊢ ∅:∅–1-1-onto→∅
11	8, 9, 10	ceqsexv2d 3447	. . . 4 ⊢ ∃𝑓 𝑓:∅–1-1-onto→∅
12		bren 8614	. . . 4 ⊢ (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅)
13	11, 12	mpbir 234	. . 3 ⊢ ∅ ≈ ∅
14		breq1 5042	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ≈ ∅ ↔ ∅ ≈ ∅))
15	13, 14	mpbiri 261	. 2 ⊢ (𝐴 = ∅ → 𝐴 ≈ ∅)
16	7, 15	impbii 212	1 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 209   = wceq 1543  ∃wex 1787  ∅c0 4223   class class class wbr 5039  ^◡ccnv 5535  –1-1-onto→wf1o 6357   ≈ cen 8601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-en 8605

This theorem is referenced by: (None)