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| Mirrors > Home > MPE Home > Th. List > ss2iun | Structured version Visualization version GIF version | ||
| Description: Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
| Ref | Expression |
|---|---|
| ss2iun | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssel 3977 | . . . . 5 ⊢ (𝐵 ⊆ 𝐶 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶)) | |
| 2 | 1 | ralimi 3083 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶)) |
| 3 | rexim 3087 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) | |
| 4 | 2, 3 | syl 17 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) |
| 5 | eliun 4995 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
| 6 | eliun 4995 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶) | |
| 7 | 4, 5, 6 | 3imtr4g 296 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)) |
| 8 | 7 | ssrdv 3989 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 ∪ ciun 4991 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-ss 3968 df-iun 4993 |
| This theorem is referenced by: iuneq2 5011 abnexg 7776 oawordri 8588 omwordri 8610 oewordri 8630 oeworde 8631 r1val1 9826 cfslb2n 10308 imasaddvallem 17574 dprdss 20049 tgcmp 23409 txcmplem1 23649 txcmplem2 23650 xkococnlem 23667 alexsubALT 24059 ptcmplem3 24062 metnrmlem2 24882 uniiccvol 25615 dvfval 25932 gsumpart 33060 bnj1145 35007 bnj1136 35011 filnetlem3 36381 poimirlem32 37659 sstotbnd2 37781 equivtotbnd 37785 trclrelexplem 43724 corcltrcl 43752 cotrclrcl 43755 ovolval5lem2 46668 ovolval5lem3 46669 smflimsuplem7 46841 |
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