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Mirrors > Home > ILE Home > Th. List > iununir | GIF version |
Description: A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33 but with biconditional changed to implication. (Contributed by Jim Kingdon, 19-Aug-2018.) |
Ref | Expression |
---|---|
iununir | ⊢ ((A ∪ ∪ B) = ∪ x ∈ B (A ∪ x) → (B = ∅ → A = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieq 3580 | . . . . . 6 ⊢ (B = ∅ → ∪ B = ∪ ∅) | |
2 | uni0 3598 | . . . . . 6 ⊢ ∪ ∅ = ∅ | |
3 | 1, 2 | syl6eq 2085 | . . . . 5 ⊢ (B = ∅ → ∪ B = ∅) |
4 | 3 | uneq2d 3091 | . . . 4 ⊢ (B = ∅ → (A ∪ ∪ B) = (A ∪ ∅)) |
5 | un0 3245 | . . . 4 ⊢ (A ∪ ∅) = A | |
6 | 4, 5 | syl6eq 2085 | . . 3 ⊢ (B = ∅ → (A ∪ ∪ B) = A) |
7 | iuneq1 3661 | . . . 4 ⊢ (B = ∅ → ∪ x ∈ B (A ∪ x) = ∪ x ∈ ∅ (A ∪ x)) | |
8 | 0iun 3705 | . . . 4 ⊢ ∪ x ∈ ∅ (A ∪ x) = ∅ | |
9 | 7, 8 | syl6eq 2085 | . . 3 ⊢ (B = ∅ → ∪ x ∈ B (A ∪ x) = ∅) |
10 | 6, 9 | eqeq12d 2051 | . 2 ⊢ (B = ∅ → ((A ∪ ∪ B) = ∪ x ∈ B (A ∪ x) ↔ A = ∅)) |
11 | 10 | biimpcd 148 | 1 ⊢ ((A ∪ ∪ B) = ∪ x ∈ B (A ∪ x) → (B = ∅ → A = ∅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∪ cun 2909 ∅c0 3218 ∪ cuni 3571 ∪ ciun 3648 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-sn 3373 df-uni 3572 df-iun 3650 |
This theorem is referenced by: (None) |
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