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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssunib | Structured version Visualization version GIF version |
Description: Two ways to say a class is a subclass of a union. (Contributed by RP, 27-Jan-2025.) |
Ref | Expression |
---|---|
ssunib | ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss3 3965 | . 2 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵) | |
2 | eluni2 4906 | . . 3 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) | |
3 | 2 | ralbii 3087 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) |
4 | 1, 3 | bitri 275 | 1 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 ⊆ wss 3943 ∪ cuni 4902 |
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 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-v 3470 df-in 3950 df-ss 3960 df-uni 4903 |
This theorem is referenced by: onmaxnelsup 42529 onsupnmax 42534 |
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