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| Description: Refinement is a relation. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.) | 
| Ref | Expression | 
|---|---|
| refrel | ⊢ Rel Ref | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-ref 23513 | . 2 ⊢ Ref = {〈𝑥, 𝑦〉 ∣ (∪ 𝑦 = ∪ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)} | |
| 2 | 1 | relopabiv 5830 | 1 ⊢ Rel Ref | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1540 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 ∪ cuni 4907 Rel wrel 5690 Refcref 23510 | 
| 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-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-opab 5206 df-xp 5691 df-rel 5692 df-ref 23513 | 
| This theorem is referenced by: isref 23517 refbas 23518 refssex 23519 reftr 23522 refun0 23523 locfinref 33840 refssfne 36359 | 
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