Theorem en0r 9017

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 8951	. . . . 5 ⊢ (∅ ≈ 𝐴 → (∅ ∈ V ∧ 𝐴 ∈ V))
2		breng 8952	. . . . 5 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ≈ 𝐴 ↔ ∃𝑓 𝑓:∅–1-1-onto→𝐴))
3	1, 2	syl 18	. . . 4 ⊢ (∅ ≈ 𝐴 → (∅ ≈ 𝐴 ↔ ∃𝑓 𝑓:∅–1-1-onto→𝐴))
4	3	ibi 270	. . 3 ⊢ (∅ ≈ 𝐴 → ∃𝑓 𝑓:∅–1-1-onto→𝐴)
5		f1o00 6857	. . . . 5 ⊢ (𝑓:∅–1-1-onto→𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
6	5	simprbi 502	. . . 4 ⊢ (𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅)
7	6	exlimiv 1957	. . 3 ⊢ (∃𝑓 𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅)
8	4, 7	syl 18	. 2 ⊢ (∅ ≈ 𝐴 → 𝐴 = ∅)
9		0ex 5272	. . . . 5 ⊢ ∅ ∈ V
10		f1oeq1 6809	. . . . 5 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
11		f1o0 6859	. . . . 5 ⊢ ∅:∅–1-1-onto→∅
12	9, 10, 11	ceqsexv2d 3512	. . . 4 ⊢ ∃𝑓 𝑓:∅–1-1-onto→∅
13		breng 8952	. . . . 5 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅))
14	9, 9, 13	mp2an 704	. . . 4 ⊢ (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅)
15	12, 14	mpbir 234	. . 3 ⊢ ∅ ≈ ∅
16		breq2 5117	. . 3 ⊢ (𝐴 = ∅ → (∅ ≈ 𝐴 ↔ ∅ ≈ ∅))
17	15, 16	mpbiri 261	. 2 ⊢ (𝐴 = ∅ → ∅ ≈ 𝐴)
18	8, 17	impbii 212	1 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  Vcvv 3463  ∅c0 4294   class class class wbr 5113  –1-1-onto→wf1o 6536   ≈ cen 8940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-mo 2573  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-en 8944

This theorem is referenced by:  0sdomg  9094  fiint  9286  rp-isfinite6  44170  ensucne0  44181