Theorem en0r 8968

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 8903	. . . . 5 ⊢ (∅ ≈ 𝐴 → (∅ ∈ V ∧ 𝐴 ∈ V))
2		breng 8904	. . . . 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 6817	. . . . 5 ⊢ (𝑓:∅–1-1-onto→𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
6	5	simprbi 496	. . . 4 ⊢ (𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅)
7	6	exlimiv 1930	. . 3 ⊢ (∃𝑓 𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅)
8	4, 7	syl 17	. 2 ⊢ (∅ ≈ 𝐴 → 𝐴 = ∅)
9		0ex 5257	. . . . 5 ⊢ ∅ ∈ V
10		f1oeq1 6770	. . . . 5 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
11		f1o0 6819	. . . . 5 ⊢ ∅:∅–1-1-onto→∅
12	9, 10, 11	ceqsexv2d 3496	. . . 4 ⊢ ∃𝑓 𝑓:∅–1-1-onto→∅
13		breng 8904	. . . . 5 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅))
14	9, 9, 13	mp2an 692	. . . 4 ⊢ (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅)
15	12, 14	mpbir 231	. . 3 ⊢ ∅ ≈ ∅
16		breq2 5106	. . 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 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3444  ∅c0 4292   class class class wbr 5102  –1-1-onto→wf1o 6498   ≈ cen 8892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-en 8896

This theorem is referenced by:  0sdomg  9047  fiint  9253  rp-isfinite6  43500  ensucne0  43511