![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
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 3962 | . 2 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵) | |
2 | eluni2 4903 | . . 3 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) | |
3 | 2 | ralbii 3085 | . 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 3053 ∃wrex 3062 ⊆ wss 3940 ∪ cuni 4899 |
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 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-v 3468 df-in 3947 df-ss 3957 df-uni 4900 |
This theorem is referenced by: onmaxnelsup 42427 onsupnmax 42432 |
Copyright terms: Public domain | W3C validator |