Theorem en0OLD 8621

Description: Obsolete version of en0 8620 as of 31-Jul-2024. (Contributed by NM, 27-May-1998.) (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 8566	. . 3 ⊢ (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴–1-1-onto→∅)
2		f1ocnv 6632	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→∅ → ◡𝑓:∅–1-1-onto→𝐴)
3		f1o00 6654	. . . . . 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 1937	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅)
7	1, 6	sylbi 220	. 2 ⊢ (𝐴 ≈ ∅ → 𝐴 = ∅)
8		0ex 5175	. . . 4 ⊢ ∅ ∈ V
9	8	enref 8590	. . 3 ⊢ ∅ ≈ ∅
10		breq1 5033	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ≈ ∅ ↔ ∅ ≈ ∅))
11	9, 10	mpbiri 261	. 2 ⊢ (𝐴 = ∅ → 𝐴 ≈ ∅)
12	7, 11	impbii 212	1 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 209   = wceq 1542  ∃wex 1786  ∅c0 4211   class class class wbr 5030  ^◡ccnv 5524  –1-1-onto→wf1o 6338   ≈ cen 8554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-en 8558

This theorem is referenced by: (None)