Theorem n0rf 3447

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 3448 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 1512	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
2		n0rf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3		nfcv 2329	. . . . 5 ⊢ Ⅎ𝑥∅
4	2, 3	cleqf 2354	. . . 4 ⊢ (𝐴 = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
5		noel 3438	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
6	5	nbn 700	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
7	6	albii 1480	. . . 4 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅))
8	4, 7	bitr4i 187	. . 3 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
9	8	necon3abii 2393	. 2 ⊢ (𝐴 ≠ ∅ ↔ ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
10	1, 9	sylibr 134	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  ∀wal 1361   = wceq 1363  ∃wex 1502   ∈ wcel 2158  Ⅎwnfc 2316   ≠ wne 2357  ∅c0 3434

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-v 2751  df-dif 3143  df-nul 3435

This theorem is referenced by:  n0r  3448  abn0r  3459