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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rp-intrabeq | Structured version Visualization version GIF version | ||
| Description: Equality theorem for supremum of sets of ordinals. (Contributed by RP, 23-Jan-2025.) |
| Ref | Expression |
|---|---|
| rp-intrabeq | ⊢ (𝐴 = 𝐵 → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | raleq 3323 | . . 3 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)) | |
| 2 | 1 | rabbidv 3444 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} = {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥}) |
| 3 | 2 | inteqd 4951 | 1 ⊢ (𝐴 = 𝐵 → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∀wral 3061 {crab 3436 ⊆ wss 3951 ∩ cint 4946 Oncon0 6384 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-ral 3062 df-rex 3071 df-rab 3437 df-int 4947 |
| This theorem is referenced by: (None) |
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