Theorem en0 9073

Description: The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5331, ax-un 7775. (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 9009	. . . . 5 ⊢ (𝐴 ≈ ∅ → (𝐴 ∈ V ∧ ∅ ∈ V))
2		breng 9010	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴⟶1-1onto∅))
3	1, 2	syl 18	. . . 4 ⊢ (𝐴 ≈ ∅ → (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴⟶1-1onto∅))
4	3	ibi 270	. . 3 ⊢ (𝐴 ≈ ∅ → ∃𝑓 𝑓:𝐴⟶1-1onto∅)
5		f1ocnv 6859	. . . . 5 ⊢ (𝑓:𝐴⟶1-1onto∅ → ◡𝑓:∅⟶1-1onto𝐴)
6		f1o00 6883	. . . . . 6 ⊢ (◡𝑓:∅⟶1-1onto𝐴 ↔ (◡𝑓 = ∅ ∧ 𝐴 = ∅))
7	6	simprbi 503	. . . . 5 ⊢ (◡𝑓:∅⟶1-1onto𝐴 → 𝐴 = ∅)
8	5, 7	syl 18	. . . 4 ⊢ (𝑓:𝐴⟶1-1onto∅ → 𝐴 = ∅)
9	8	exlimiv 1965	. . 3 ⊢ (∃𝑓 𝑓:𝐴⟶1-1onto∅ → 𝐴 = ∅)
10	4, 9	syl 18	. 2 ⊢ (𝐴 ≈ ∅ → 𝐴 = ∅)
11		0ex 5264	. . . . 5 ⊢ ∅ ∈ V
12		f1oeq1 6834	. . . . 5 ⊢ (𝑓 = ∅ → (𝑓:∅⟶1-1onto∅ ↔ ∅:∅⟶1-1onto∅))
13		f1o0 6885	. . . . 5 ⊢ ∅:∅⟶1-1onto∅
14	11, 12, 13	ceqsexv2d 3503	. . . 4 ⊢ ∃𝑓 𝑓:∅⟶1-1onto∅
15		breng 9010	. . . . 5 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅⟶1-1onto∅))
16	11, 11, 15	mp2an 705	. . . 4 ⊢ (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅⟶1-1onto∅)
17	14, 16	mpbir 234	. . 3 ⊢ ∅ ≈ ∅
18		breq1 5109	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ≈ ∅ ↔ ∅ ≈ ∅))
19	17, 18	mpbiri 261	. 2 ⊢ (𝐴 = ∅ → 𝐴 ≈ ∅)
20	10, 19	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  ^◡ccnv 5654  ⟶1-1ontowf1o 6557   ≈ cen 8998

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 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-en 9002

This theorem is used by:  0fi  9098  enrefnn  9102  dom0  9152  sdom0  9156  findcard  9207  findcard2  9208  nneneq  9249  cantnff  9702  cantnf0  9703  cantnfp1lem2  9707  cantnflem1  9717  cantnf  9721  cnfcom2lem  9729  cardnueq0  10072  infmap2  10322  fin23lem26  10430  cardeq0  10663  hasheq0  14534  mreexexd  17851  pmtrfmvdn0  19706  pmtrsn  19763  kard0  35922  kard0b  35927  rp-isfinite6  44711  ensucne0OLD  44723