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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 2733 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝐴 | |
2 | ssid 4005 | . . . . 5 ⊢ 𝑥 ⊆ 𝑥 | |
3 | sseq2 4009 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑥)) | |
4 | 3 | rspcev 3613 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ⊆ 𝑥) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) |
5 | 2, 4 | mpan2 690 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) |
6 | 5 | rgen 3064 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦 |
7 | 1, 6 | pm3.2i 472 | . 2 ⊢ (∪ 𝐴 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦) |
8 | 1, 1 | isref 23013 | . 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 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ⊆ wss 3949 ∪ cuni 4909 class class class wbr 5149 Refcref 23006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-ref 23009 |
This theorem is referenced by: locfinref 32821 |
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