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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 3304 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
2 | 1 | ex 415 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) |
3 | eliun 4923 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
4 | 2, 3 | syl6ibr 254 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)) |
5 | 4 | ssrdv 3973 | 1 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∃wrex 3139 ⊆ wss 3936 ∪ ciun 4919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-v 3496 df-in 3943 df-ss 3952 df-iun 4921 |
This theorem is referenced by: ssiun2s 4972 disjxiun 5063 triun 5185 iunopeqop 5411 ixpf 8484 ixpiunwdom 9055 r1sdom 9203 r1val1 9215 rankuni2b 9282 rankval4 9296 cplem1 9318 domtriomlem 9864 ac6num 9901 iunfo 9961 iundom2g 9962 pwfseqlem3 10082 inar1 10197 tskuni 10205 iunconnlem 22035 ptclsg 22223 ovoliunlem1 24103 limciun 24492 ssiun2sf 30311 suppovss 30426 bnj906 32202 bnj999 32230 bnj1014 32233 bnj1408 32308 trpredrec 33077 rdgssun 34662 cpcolld 40614 iunmapss 41498 ssmapsn 41499 sge0iunmpt 42720 sge0iun 42721 voliunsge0lem 42774 omeiunltfirp 42821 |
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