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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 3320 | . . 3 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)) | |
2 | 1 | rabbidv 3440 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} = {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥}) |
3 | 2 | inteqd 4955 | 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 1536 ∀wral 3058 {crab 3432 ⊆ wss 3962 ∩ cint 4950 Oncon0 6385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-ral 3059 df-rex 3068 df-rab 3433 df-int 4951 |
This theorem is referenced by: (None) |
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