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Mirrors > Home > MPE Home > Th. List > iunss1 | Structured version Visualization version GIF version |
Description: Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
iunss1 | ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐵 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrexv 3630 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 → ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)) | |
2 | eliun 4460 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶) | |
3 | eliun 4460 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶) | |
4 | 1, 2, 3 | 3imtr4g 284 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶)) |
5 | 4 | ssrdv 3574 | 1 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐵 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∃wrex 2897 ⊆ wss 3540 ∪ ciun 4455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-in 3547 df-ss 3554 df-iun 4457 |
This theorem is referenced by: iuneq1 4470 iunxdif2 4504 oelim2 7562 fsumiun 14394 ovolfiniun 23076 uniioovol 23153 usgreghash2spotv 26593 esum2dlem 29481 esum2d 29482 carsgclctunlem2 29708 bnj1413 30357 bnj1408 30358 volsupnfl 32624 corclrcl 37018 cotrcltrcl 37036 iuneqfzuzlem 38491 fsumiunss 38642 sge0iunmptlemfi 39306 sge0iunmptlemre 39308 carageniuncllem1 39411 carageniuncllem2 39412 caratheodorylem2 39417 ovnsubaddlem1 39460 fusgreghash2wspv 41499 |
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