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Mirrors > Home > NFE Home > Th. List > prnzg | GIF version |
Description: A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.) |
Ref | Expression |
---|---|
prnzg | ⊢ (A ∈ V → {A, B} ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq1 3799 | . . 3 ⊢ (x = A → {x, B} = {A, B}) | |
2 | 1 | neeq1d 2529 | . 2 ⊢ (x = A → ({x, B} ≠ ∅ ↔ {A, B} ≠ ∅)) |
3 | vex 2862 | . . 3 ⊢ x ∈ V | |
4 | 3 | prnz 3835 | . 2 ⊢ {x, B} ≠ ∅ |
5 | 2, 4 | vtoclg 2914 | 1 ⊢ (A ∈ V → {A, B} ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 ∅c0 3550 {cpr 3738 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-nul 3551 df-sn 3741 df-pr 3742 |
This theorem is referenced by: (None) |
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