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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 2732 | . . 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 689 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) |
6 | 5 | rgen 3063 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦 |
7 | 1, 6 | pm3.2i 471 | . 2 ⊢ (∪ 𝐴 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) |
8 | 1, 1 | isref 23012 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴Ref𝐴 ↔ (∪ 𝐴 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦))) |
9 | 7, 8 | mpbiri 257 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴Ref𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 ⊆ wss 3948 ∪ cuni 4908 class class class wbr 5148 Refcref 23005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 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 23008 |
This theorem is referenced by: locfinref 32816 |
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