![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2734 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝐴 | |
2 | ssid 4025 | . . . . 5 ⊢ 𝑥 ⊆ 𝑥 | |
3 | sseq2 4029 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑥)) | |
4 | 3 | rspcev 3631 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑥) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) |
5 | 2, 4 | mpan2 690 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) |
6 | 5 | rgen 3065 | . . 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 23531 | . 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 1537 ∈ wcel 2103 ∀wral 3063 ∃wrex 3072 ⊆ wss 3970 ∪ cuni 4931 class class class wbr 5169 Refcref 23524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-xp 5705 df-rel 5706 df-ref 23527 |
This theorem is referenced by: locfinref 33779 |
Copyright terms: Public domain | W3C validator |