Theorem n0rf 3427

Description: An inhabited class is nonempty. Following the Definition of [Bauer], p. 483, we call a class 𝐴 nonempty if 𝐴 ≠ ∅ and inhabited if it has at least one element. In classical logic these two concepts are equivalent, for example see Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0r 3428 requires only that 𝑥 not be free in, rather than not occur in, 𝐴. (Contributed by Jim Kingdon, 31-Jul-2018.)

Hypothesis

Ref	Expression
n0rf.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
n0rf	⊢ (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)

Proof of Theorem n0rf

Step	Hyp	Ref	Expression
1		exalim 1495	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
2		n0rf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3		nfcv 2312	. . . . 5 ⊢ Ⅎ𝑥∅
4	2, 3	cleqf 2337	. . . 4 ⊢ (𝐴 = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
5		noel 3418	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
6	5	nbn 694	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
7	6	albii 1463	. . . 4 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
8	4, 7	bitr4i 186	. . 3 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
9	8	necon3abii 2376	. 2 ⊢ (𝐴 ≠ ∅ ↔ ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
10	1, 9	sylibr 133	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  ∀wal 1346   = wceq 1348  ∃wex 1485   ∈ wcel 2141  Ⅎwnfc 2299   ≠ wne 2340  ∅c0 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-v 2732  df-dif 3123  df-nul 3415

This theorem is referenced by:  n0r  3428  abn0r  3439