Theorem notm0 3515

Description: A class is not inhabited if and only if it is empty. (Contributed by Jim Kingdon, 1-Jul-2022.)

Assertion

Ref	Expression
notm0	⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 = ∅)

Distinct variable group:   𝑥,𝐴

Proof of Theorem notm0

Step	Hyp	Ref	Expression
1		eq0 3513	. 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
2		alnex 1547	. 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ¬ ∃𝑥 𝑥 ∈ 𝐴)
3	1, 2	bitr2i 185	1 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 = ∅)

Syntax hints:  ¬ wn 3   ↔ wb 105  ∀wal 1395   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∅c0 3494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-nul 3495

This theorem is referenced by:  disjnim  4078  pwntru  4289  exmidn0m  4291  mapprc  6821  map0g  6857  ixpprc  6888  ixp0  6900  exmidfodomrlemim  7412  ntreq0  14862  blssioo  15283  lgsquadlem3  15814  pw0ss  15940  g0wlk0  16227  pwtrufal  16624