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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssiun3 | Structured version Visualization version GIF version |
Description: Subset equivalence for an indexed union. (Contributed by Thierry Arnoux, 17-Oct-2016.) |
Ref | Expression |
---|---|
ssiun3 | ⊢ (∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 3967 | . 2 ⊢ (𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)) | |
2 | df-ral 3058 | . 2 ⊢ (∀𝑦 ∈ 𝐶 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐶 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)) | |
3 | eliun 5002 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
4 | 3 | ralbii 3089 | . 2 ⊢ (∀𝑦 ∈ 𝐶 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) |
5 | 1, 2, 4 | 3bitr2ri 299 | 1 ⊢ (∀𝑦 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1531 ∈ wcel 2098 ∀wral 3057 ∃wrex 3066 ⊆ wss 3947 ∪ ciun 4998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3058 df-rex 3067 df-v 3473 df-in 3954 df-ss 3964 df-iun 5000 |
This theorem is referenced by: (None) |
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