Theorem n0rf 3459

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 3460 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 1513	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
2		n0rf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3		nfcv 2336	. . . . 5 ⊢ Ⅎ𝑥∅
4	2, 3	cleqf 2361	. . . 4 ⊢ (𝐴 = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
5		noel 3450	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
6	5	nbn 700	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
7	6	albii 1481	. . . 4 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
8	4, 7	bitr4i 187	. . 3 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
9	8	necon3abii 2400	. 2 ⊢ (𝐴 ≠ ∅ ↔ ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
10	1, 9	sylibr 134	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  ∀wal 1362   = wceq 1364  ∃wex 1503   ∈ wcel 2164  Ⅎwnfc 2323   ≠ wne 2364  ∅c0 3446

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-v 2762  df-dif 3155  df-nul 3447

This theorem is referenced by:  n0r  3460  abn0r  3471