Theorem en0r 9003

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 8937	. . . . 5 ⊢ (∅ ≈ 𝐴 → (∅ ∈ V ∧ 𝐴 ∈ V))
2		breng 8938	. . . . 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 269	. . 3 ⊢ (∅ ≈ 𝐴 → ∃𝑓 𝑓:∅–1-1-onto→𝐴)
5		f1o00 6844	. . . . 5 ⊢ (𝑓:∅–1-1-onto→𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
6	5	simprbi 501	. . . 4 ⊢ (𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅)
7	6	exlimiv 1952	. . 3 ⊢ (∃𝑓 𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅)
8	4, 7	syl 17	. 2 ⊢ (∅ ≈ 𝐴 → 𝐴 = ∅)
9		0ex 5259	. . . . 5 ⊢ ∅ ∈ V
10		f1oeq1 6796	. . . . 5 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
11		f1o0 6846	. . . . 5 ⊢ ∅:∅–1-1-onto→∅
12	9, 10, 11	ceqsexv2d 3505	. . . 4 ⊢ ∃𝑓 𝑓:∅–1-1-onto→∅
13		breng 8938	. . . . 5 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅))
14	9, 9, 13	mp2an 702	. . . 4 ⊢ (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅)
15	12, 14	mpbir 233	. . 3 ⊢ ∅ ≈ ∅
16		breq2 5106	. . 3 ⊢ (𝐴 = ∅ → (∅ ≈ 𝐴 ↔ ∅ ≈ ∅))
17	15, 16	mpbiri 260	. 2 ⊢ (𝐴 = ∅ → ∅ ≈ 𝐴)
18	8, 17	impbii 211	1 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1562  ∃wex 1801   ∈ wcel 2144  Vcvv 3456  ∅c0 4287   class class class wbr 5102  –1-1-onto→wf1o 6522   ≈ cen 8926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-mo 2568  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-en 8930

This theorem is referenced by:  0sdomg  9080  fiint  9273  rp-isfinite6  44099  ensucne0  44110