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Mirrors > Home > MPE Home > Th. List > ssiun | Structured version Visualization version GIF version |
Description: Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
ssiun | ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3880 | . . . . 5 ⊢ (𝐶 ⊆ 𝐵 → (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵)) | |
2 | 1 | reximi 3156 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵)) |
3 | r19.37v 3248 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵) → (𝑦 ∈ 𝐶 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → (𝑦 ∈ 𝐶 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) |
5 | eliun 4894 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
6 | 4, 5 | syl6ibr 255 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → (𝑦 ∈ 𝐶 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)) |
7 | 6 | ssrdv 3893 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 ∃wrex 3052 ⊆ wss 3853 ∪ ciun 4890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-v 3400 df-in 3860 df-ss 3870 df-iun 4892 |
This theorem is referenced by: iunss2 4944 iunpwss 5001 iunpw 7534 wfrdmcl 8041 onfununi 8056 oen0 8292 trpredtr 9313 dftrpred3g 9317 trcl 9322 rtrclreclem1 14585 rtrclreclem2 14587 |
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