Theorem en0r 9083

Description: The empty set is equinumerous only to itself. (Contributed by BTernaryTau, 29-Nov-2024.)

Assertion

Ref	Expression
en0r	⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅)

Proof of Theorem en0r

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		encv 9013	. . . . 5 ⊢ (∅ ≈ 𝐴 → (∅ ∈ V ∧ 𝐴 ∈ V))
2		breng 9014	. . . . 5 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ≈ 𝐴 ↔ ∃𝑓 𝑓:∅–1-1-onto→𝐴))
3	1, 2	syl 17	. . . 4 ⊢ (∅ ≈ 𝐴 → (∅ ≈ 𝐴 ↔ ∃𝑓 𝑓:∅–1-1-onto→𝐴))
4	3	ibi 267	. . 3 ⊢ (∅ ≈ 𝐴 → ∃𝑓 𝑓:∅–1-1-onto→𝐴)
5		f1o00 6899	. . . . 5 ⊢ (𝑓:∅–1-1-onto→𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
6	5	simprbi 496	. . . 4 ⊢ (𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅)
7	6	exlimiv 1929	. . 3 ⊢ (∃𝑓 𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅)
8	4, 7	syl 17	. 2 ⊢ (∅ ≈ 𝐴 → 𝐴 = ∅)
9		0ex 5325	. . . . 5 ⊢ ∅ ∈ V
10		f1oeq1 6852	. . . . 5 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
11		f1o0 6901	. . . . 5 ⊢ ∅:∅–1-1-onto→∅
12	9, 10, 11	ceqsexv2d 3545	. . . 4 ⊢ ∃𝑓 𝑓:∅–1-1-onto→∅
13		breng 9014	. . . . 5 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅))
14	9, 9, 13	mp2an 691	. . . 4 ⊢ (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅)
15	12, 14	mpbir 231	. . 3 ⊢ ∅ ≈ ∅
16		breq2 5170	. . 3 ⊢ (𝐴 = ∅ → (∅ ≈ 𝐴 ↔ ∅ ≈ ∅))
17	15, 16	mpbiri 258	. 2 ⊢ (𝐴 = ∅ → ∅ ≈ 𝐴)
18	8, 17	impbii 209	1 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  Vcvv 3488  ∅c0 4352   class class class wbr 5166  –1-1-onto→wf1o 6574   ≈ cen 9002

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 2110  ax-9 2118  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

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 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-en 9006

This theorem is referenced by:  0sdomg  9172  fiint  9396  rp-isfinite6  43482  ensucne0  43493