Theorem en0r 9072

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 9006	. . . . 5 ⊢ (∅ ≈ 𝐴 → (∅ ∈ V ∧ 𝐴 ∈ V))
2		breng 9007	. . . . 5 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ≈ 𝐴 ↔ ∃𝑓 𝑓:∅⟶1-1onto𝐴))
3	1, 2	syl 18	. . . 4 ⊢ (∅ ≈ 𝐴 → (∅ ≈ 𝐴 ↔ ∃𝑓 𝑓:∅⟶1-1onto𝐴))
4	3	ibi 270	. . 3 ⊢ (∅ ≈ 𝐴 → ∃𝑓 𝑓:∅⟶1-1onto𝐴)
5		f1o00 6880	. . . . 5 ⊢ (𝑓:∅⟶1-1onto𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅))
6	5	simprbi 503	. . . 4 ⊢ (𝑓:∅⟶1-1onto𝐴 → 𝐴 = ∅)
7	6	exlimiv 1965	. . 3 ⊢ (∃𝑓 𝑓:∅⟶1-1onto𝐴 → 𝐴 = ∅)
8	4, 7	syl 18	. 2 ⊢ (∅ ≈ 𝐴 → 𝐴 = ∅)
9		0ex 5264	. . . . 5 ⊢ ∅ ∈ V
10		f1oeq1 6831	. . . . 5 ⊢ (𝑓 = ∅ → (𝑓:∅⟶1-1onto∅ ↔ ∅:∅⟶1-1onto∅))
11		f1o0 6882	. . . . 5 ⊢ ∅:∅⟶1-1onto∅
12	9, 10, 11	ceqsexv2d 3503	. . . 4 ⊢ ∃𝑓 𝑓:∅⟶1-1onto∅
13		breng 9007	. . . . 5 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅⟶1-1onto∅))
14	9, 9, 13	mp2an 705	. . . 4 ⊢ (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅⟶1-1onto∅)
15	12, 14	mpbir 234	. . 3 ⊢ ∅ ≈ ∅
16		breq2 5110	. . 3 ⊢ (𝐴 = ∅ → (∅ ≈ 𝐴 ↔ ∅ ≈ ∅))
17	15, 16	mpbiri 261	. 2 ⊢ (𝐴 = ∅ → ∅ ≈ 𝐴)
18	8, 17	impbii 212	1 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454  ∅c0 4281   class class class wbr 5106  ⟶1-1ontowf1o 6554   ≈ cen 8995

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-en 8999

This theorem is used by:  0sdomg  9150  fiint  9343  rp-isfinite6  44709  ensucne0  44720