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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 3968 | . 2 ⊢ (𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵) | |
2 | eluni2 4912 | . . 3 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) | |
3 | 2 | ralbii 3090 | . 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 2099 ∀wral 3058 ∃wrex 3067 ⊆ wss 3947 ∪ cuni 4908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-v 3473 df-in 3954 df-ss 3964 df-uni 4909 |
This theorem is referenced by: onmaxnelsup 42651 onsupnmax 42656 |
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