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Mirrors > Home > ILE Home > Th. List > iuncom | GIF version |
Description: Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.) |
Ref | Expression |
---|---|
iuncom | ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = ∪ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom 2641 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶) | |
2 | eliun 3890 | . . . . 5 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶) | |
3 | 2 | rexbii 2484 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ∪ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶) |
4 | eliun 3890 | . . . . 5 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶) | |
5 | 4 | rexbii 2484 | . . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶) |
6 | 1, 3, 5 | 3bitr4i 212 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ∪ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶) |
7 | eliun 3890 | . . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ∪ 𝑦 ∈ 𝐵 𝐶) | |
8 | eliun 3890 | . . 3 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶) | |
9 | 6, 7, 8 | 3bitr4i 212 | . 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 ↔ 𝑧 ∈ ∪ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶) |
10 | 9 | eqriv 2174 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = ∪ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 ∃wrex 2456 ∪ ciun 3886 |
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-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-iun 3888 |
This theorem is referenced by: (None) |
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