Theorem en0 8935

Description: The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5301, ax-un 7663. (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 8872	. . . . 5 ⊢ (𝐴 ≈ ∅ → (𝐴 ∈ V ∧ ∅ ∈ V))
2		breng 8873	. . . . 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 6771	. . . . 5 ⊢ (𝑓:𝐴–1-1-onto→∅ → ◡𝑓:∅–1-1-onto→𝐴)
6		f1o00 6794	. . . . . 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 1931	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅)
10	4, 9	syl 17	. 2 ⊢ (𝐴 ≈ ∅ → 𝐴 = ∅)
11		0ex 5243	. . . . 5 ⊢ ∅ ∈ V
12		f1oeq1 6747	. . . . 5 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
13		f1o0 6796	. . . . 5 ⊢ ∅:∅–1-1-onto→∅
14	11, 12, 13	ceqsexv2d 3486	. . . 4 ⊢ ∃𝑓 𝑓:∅–1-1-onto→∅
15		breng 8873	. . . . 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 5092	. . 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 1541  ∃wex 1780   ∈ wcel 2110  Vcvv 3434  ∅c0 4281   class class class wbr 5089  ^◡ccnv 5613  –1-1-onto→wf1o 6476   ≈ cen 8861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-mo 2534  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-en 8865

This theorem is referenced by:  0fi  8959  enrefnn  8963  dom0  9013  sdom0  9017  findcard  9068  findcard2  9069  nneneq  9110  cantnff  9559  cantnf0  9560  cantnfp1lem2  9564  cantnflem1  9574  cantnf  9578  cnfcom2lem  9586  cardnueq0  9849  infmap2  10100  fin23lem26  10208  cardeq0  10435  hasheq0  14262  mreexexd  17546  pmtrfmvdn0  19367  pmtrsn  19424  rp-isfinite6  43530  ensucne0OLD  43542