Theorem en0 8991

Description: The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5322, ax-un 7713. (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 8928	. . . . 5 ⊢ (𝐴 ≈ ∅ → (𝐴 ∈ V ∧ ∅ ∈ V))
2		breng 8929	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴–1-1-onto→∅))
3	1, 2	syl 17	. . . 4 ⊢ (𝐴 ≈ ∅ → (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴–1-1-onto→∅))
4	3	ibi 267	. . 3 ⊢ (𝐴 ≈ ∅ → ∃𝑓 𝑓:𝐴–1-1-onto→∅)
5		f1ocnv 6814	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→∅ → ◡𝑓:∅–1-1-onto→𝐴)
6		f1o00 6837	. . . . . 6 ⊢ (◡𝑓:∅–1-1-onto→𝐴 ↔ (◡𝑓 = ∅ ∧ 𝐴 = ∅))
7	6	simprbi 496	. . . . 5 ⊢ (◡𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅)
8	5, 7	syl 17	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅)
9	8	exlimiv 1930	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅)
10	4, 9	syl 17	. 2 ⊢ (𝐴 ≈ ∅ → 𝐴 = ∅)
11		0ex 5264	. . . . 5 ⊢ ∅ ∈ V
12		f1oeq1 6790	. . . . 5 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
13		f1o0 6839	. . . . 5 ⊢ ∅:∅–1-1-onto→∅
14	11, 12, 13	ceqsexv2d 3502	. . . 4 ⊢ ∃𝑓 𝑓:∅–1-1-onto→∅
15		breng 8929	. . . . 5 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅))
16	11, 11, 15	mp2an 692	. . . 4 ⊢ (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅)
17	14, 16	mpbir 231	. . 3 ⊢ ∅ ≈ ∅
18		breq1 5112	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ≈ ∅ ↔ ∅ ≈ ∅))
19	17, 18	mpbiri 258	. 2 ⊢ (𝐴 = ∅ → 𝐴 ≈ ∅)
20	10, 19	impbii 209	1 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3450  ∅c0 4298   class class class wbr 5109  ^◡ccnv 5639  –1-1-onto→wf1o 6512   ≈ cen 8917

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 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

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 2534  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-br 5110  df-opab 5172  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-en 8921

This theorem is referenced by:  0fi  9015  snfiOLD  9017  enrefnn  9020  dom0  9074  sdom0  9078  findcard  9132  findcard2  9133  nneneq  9175  enp1iOLD  9231  fiintOLD  9284  cantnff  9633  cantnf0  9634  cantnfp1lem2  9638  cantnflem1  9648  cantnf  9652  cnfcom2lem  9660  cardnueq0  9923  infmap2  10176  fin23lem26  10284  cardeq0  10511  hasheq0  14334  mreexexd  17615  pmtrfmvdn0  19398  pmtrsn  19455  rp-isfinite6  43500  ensucne0OLD  43512