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Mirrors > Home > MPE Home > Th. List > ssiun2 | Structured version Visualization version GIF version |
Description: Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
ssiun2 | ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspe 2986 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
2 | 1 | ex 449 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) |
3 | eliun 4460 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
4 | 2, 3 | syl6ibr 241 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)) |
5 | 4 | ssrdv 3574 | 1 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
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: ssiun2s 4500 disjxiun 4579 disjxiunOLD 4580 triun 4694 iunopeqop 4906 ixpf 7816 ixpiunwdom 8379 r1sdom 8520 r1val1 8532 rankuni2b 8599 rankval4 8613 cplem1 8635 domtriomlem 9147 ac6num 9184 iunfo 9240 iundom2g 9241 pwfseqlem3 9361 inar1 9476 tskuni 9484 iunconlem 21040 ptclsg 21228 ovoliunlem1 23077 limciun 23464 ssiun2sf 28760 bnj906 30254 bnj999 30281 bnj1014 30284 bnj1408 30358 trpredrec 30982 iunmapss 38402 ssmapsn 38403 sge0iunmpt 39311 sge0iun 39312 voliunsge0lem 39365 omeiunltfirp 39409 |
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