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Mirrors > Home > MPE Home > Th. List > Mathboxes > unielid | Structured version Visualization version GIF version |
Description: Two ways to say the union of a class is an element of that class. (Contributed by RP, 27-Jan-2025.) |
Ref | Expression |
---|---|
unielid | ⊢ (∪ 𝐴 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 4025 | . 2 ⊢ 𝐴 ⊆ 𝐴 | |
2 | unielss 43120 | . 2 ⊢ (𝐴 ⊆ 𝐴 → (∪ 𝐴 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (∪ 𝐴 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∈ wcel 2103 ∀wral 3063 ∃wrex 3072 ⊆ wss 3970 ∪ cuni 4931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-tru 1540 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3064 df-rex 3073 df-ss 3987 df-uni 4932 |
This theorem is referenced by: onsupnmax 43130 onsupeqmax 43148 onsupeqnmax 43149 |
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