Theorem n0rf 3504

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 3505 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 1548	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
2		n0rf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3		nfcv 2372	. . . . 5 ⊢ Ⅎ𝑥∅
4	2, 3	cleqf 2397	. . . 4 ⊢ (𝐴 = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
5		noel 3495	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
6	5	nbn 704	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
7	6	albii 1516	. . . 4 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
8	4, 7	bitr4i 187	. . 3 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
9	8	necon3abii 2436	. 2 ⊢ (𝐴 ≠ ∅ ↔ ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
10	1, 9	sylibr 134	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  ∀wal 1393   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Ⅎwnfc 2359   ≠ wne 2400  ∅c0 3491

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-v 2801  df-dif 3199  df-nul 3492

This theorem is referenced by:  n0r  3505  abn0r  3516