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Mirrors > Home > MPE Home > Th. List > refref | Structured version Visualization version GIF version |
Description: Reflexivity of refinement. (Contributed by Jeff Hankins, 18-Jan-2010.) |
Ref | Expression |
---|---|
refref | ⊢ (𝐴 ∈ 𝑉 → 𝐴Ref𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝐴 | |
2 | ssid 4004 | . . . . 5 ⊢ 𝑥 ⊆ 𝑥 | |
3 | sseq2 4008 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑥)) | |
4 | 3 | rspcev 3612 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑥) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) |
5 | 2, 4 | mpan2 688 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) |
6 | 5 | rgen 3062 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦 |
7 | 1, 6 | pm3.2i 470 | . 2 ⊢ (∪ 𝐴 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) |
8 | 1, 1 | isref 23234 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴Ref𝐴 ↔ (∪ 𝐴 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))) |
9 | 7, 8 | mpbiri 258 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴Ref𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ∃wrex 3069 ⊆ wss 3948 ∪ cuni 4908 class class class wbr 5148 Refcref 23227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-ref 23230 |
This theorem is referenced by: locfinref 33120 |
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