New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > 0iun | GIF version |
Description: An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
0iun | ⊢ ∪x ∈ ∅ A = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rex0 3563 | . . . 4 ⊢ ¬ ∃x ∈ ∅ y ∈ A | |
2 | eliun 3973 | . . . 4 ⊢ (y ∈ ∪x ∈ ∅ A ↔ ∃x ∈ ∅ y ∈ A) | |
3 | 1, 2 | mtbir 290 | . . 3 ⊢ ¬ y ∈ ∪x ∈ ∅ A |
4 | noel 3554 | . . 3 ⊢ ¬ y ∈ ∅ | |
5 | 3, 4 | 2false 339 | . 2 ⊢ (y ∈ ∪x ∈ ∅ A ↔ y ∈ ∅) |
6 | 5 | eqriv 2350 | 1 ⊢ ∪x ∈ ∅ A = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ∈ wcel 1710 ∃wrex 2615 ∅c0 3550 ∪ciun 3969 |
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-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-nul 3551 df-iun 3971 |
This theorem is referenced by: iununi 4050 |
Copyright terms: Public domain | W3C validator |