Theorem en0 9014

Description: The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5336, ax-un 7732. (Revised by BTernaryTau, 23-Sep-2024.)

Assertion

Ref	Expression
en0	⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)

Proof of Theorem en0

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		encv 8950	. . . . 5 ⊢ (𝐴 ≈ ∅ → (𝐴 ∈ V ∧ ∅ ∈ V))
2		breng 8951	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴–1-1-onto→∅))
3	1, 2	syl 18	. . . 4 ⊢ (𝐴 ≈ ∅ → (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴–1-1-onto→∅))
4	3	ibi 270	. . 3 ⊢ (𝐴 ≈ ∅ → ∃𝑓 𝑓:𝐴–1-1-onto→∅)
5		f1ocnv 6833	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→∅ → ◡𝑓:∅–1-1-onto→𝐴)
6		f1o00 6856	. . . . . 6 ⊢ (◡𝑓:∅–1-1-onto→𝐴 ↔ (◡𝑓 = ∅ ∧ 𝐴 = ∅))
7	6	simprbi 502	. . . . 5 ⊢ (◡𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅)
8	5, 7	syl 18	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅)
9	8	exlimiv 1958	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅)
10	4, 9	syl 18	. 2 ⊢ (𝐴 ≈ ∅ → 𝐴 = ∅)
11		0ex 5269	. . . . 5 ⊢ ∅ ∈ V
12		f1oeq1 6808	. . . . 5 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
13		f1o0 6858	. . . . 5 ⊢ ∅:∅–1-1-onto→∅
14	11, 12, 13	ceqsexv2d 3502	. . . 4 ⊢ ∃𝑓 𝑓:∅–1-1-onto→∅
15		breng 8951	. . . . 5 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅))
16	11, 11, 15	mp2an 704	. . . 4 ⊢ (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅)
17	14, 16	mpbir 234	. . 3 ⊢ ∅ ≈ ∅
18		breq1 5111	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ≈ ∅ ↔ ∅ ≈ ∅))
19	17, 18	mpbiri 261	. 2 ⊢ (𝐴 = ∅ → 𝐴 ≈ ∅)
20	10, 19	impbii 212	1 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1568  ∃wex 1807   ∈ wcel 2141  Vcvv 3453  ∅c0 4285   class class class wbr 5108  ^◡ccnv 5660  –1-1-onto→wf1o 6535   ≈ cen 8939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-en 8943

This theorem is referenced by:  0fi  9038  enrefnn  9042  dom0  9092  sdom0  9096  findcard  9147  findcard2  9148  nneneq  9189  cantnff  9642  cantnf0  9643  cantnfp1lem2  9647  cantnflem1  9657  cantnf  9661  cnfcom2lem  9669  cardnueq0  9949  infmap2  10199  fin23lem26  10308  cardeq0  10535  hasheq0  14398  mreexexd  17703  pmtrfmvdn0  19531  pmtrsn  19588  kard0  35533  kard0b  35538  rp-isfinite6  44214  ensucne0OLD  44226