Theorem n0rf 3507

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 3508 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 1550	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
2		n0rf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3		nfcv 2374	. . . . 5 ⊢ Ⅎ𝑥∅
4	2, 3	cleqf 2399	. . . 4 ⊢ (𝐴 = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
5		noel 3498	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
6	5	nbn 706	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
7	6	albii 1518	. . . 4 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
8	4, 7	bitr4i 187	. . 3 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
9	8	necon3abii 2438	. 2 ⊢ (𝐴 ≠ ∅ ↔ ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
10	1, 9	sylibr 134	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  ∀wal 1395   = wceq 1397  ∃wex 1540   ∈ wcel 2202  Ⅎwnfc 2361   ≠ wne 2402  ∅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-ne 2403  df-v 2804  df-dif 3202  df-nul 3495

This theorem is referenced by:  n0r  3508  abn0r  3519