New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > un00 | GIF version |
Description: Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.) |
Ref | Expression |
---|---|
un00 | ⊢ ((A = ∅ ∧ B = ∅) ↔ (A ∪ B) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq12 3413 | . . 3 ⊢ ((A = ∅ ∧ B = ∅) → (A ∪ B) = (∅ ∪ ∅)) | |
2 | un0 3575 | . . 3 ⊢ (∅ ∪ ∅) = ∅ | |
3 | 1, 2 | syl6eq 2401 | . 2 ⊢ ((A = ∅ ∧ B = ∅) → (A ∪ B) = ∅) |
4 | ssun1 3426 | . . . . 5 ⊢ A ⊆ (A ∪ B) | |
5 | sseq2 3293 | . . . . 5 ⊢ ((A ∪ B) = ∅ → (A ⊆ (A ∪ B) ↔ A ⊆ ∅)) | |
6 | 4, 5 | mpbii 202 | . . . 4 ⊢ ((A ∪ B) = ∅ → A ⊆ ∅) |
7 | ss0b 3580 | . . . 4 ⊢ (A ⊆ ∅ ↔ A = ∅) | |
8 | 6, 7 | sylib 188 | . . 3 ⊢ ((A ∪ B) = ∅ → A = ∅) |
9 | ssun2 3427 | . . . . 5 ⊢ B ⊆ (A ∪ B) | |
10 | sseq2 3293 | . . . . 5 ⊢ ((A ∪ B) = ∅ → (B ⊆ (A ∪ B) ↔ B ⊆ ∅)) | |
11 | 9, 10 | mpbii 202 | . . . 4 ⊢ ((A ∪ B) = ∅ → B ⊆ ∅) |
12 | ss0b 3580 | . . . 4 ⊢ (B ⊆ ∅ ↔ B = ∅) | |
13 | 11, 12 | sylib 188 | . . 3 ⊢ ((A ∪ B) = ∅ → B = ∅) |
14 | 8, 13 | jca 518 | . 2 ⊢ ((A ∪ B) = ∅ → (A = ∅ ∧ B = ∅)) |
15 | 3, 14 | impbii 180 | 1 ⊢ ((A = ∅ ∧ B = ∅) ↔ (A ∪ B) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 = wceq 1642 ∪ cun 3207 ⊆ wss 3257 ∅c0 3550 |
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-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 |
This theorem is referenced by: undisj1 3602 undisj2 3603 0nelsuc 4400 addcass 4415 |
Copyright terms: Public domain | W3C validator |