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Mirrors > Home > MPE Home > Th. List > Mathboxes > cllem0 | Structured version Visualization version GIF version |
Description: The class of all sets with property 𝜑(𝑧) is closed under the binary operation on sets defined in 𝑅(𝑥, 𝑦). (Contributed by Richard Penner, 3-Jan-2020.) |
Ref | Expression |
---|---|
cllem0.v | ⊢ 𝑉 = {𝑧 ∣ 𝜑} |
cllem0.rex | ⊢ 𝑅 ∈ 𝑈 |
cllem0.r | ⊢ (𝑧 = 𝑅 → (𝜑 ↔ 𝜓)) |
cllem0.x | ⊢ (𝑧 = 𝑥 → (𝜑 ↔ 𝜒)) |
cllem0.y | ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜃)) |
cllem0.closed | ⊢ ((𝜒 ∧ 𝜃) → 𝜓) |
Ref | Expression |
---|---|
cllem0 | ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑅 ∈ 𝑉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cllem0.rex | . . . . . . 7 ⊢ 𝑅 ∈ 𝑈 | |
2 | 1 | elexi 3353 | . . . . . 6 ⊢ 𝑅 ∈ V |
3 | cllem0.r | . . . . . 6 ⊢ (𝑧 = 𝑅 → (𝜑 ↔ 𝜓)) | |
4 | cllem0.v | . . . . . 6 ⊢ 𝑉 = {𝑧 ∣ 𝜑} | |
5 | 2, 3, 4 | elab2 3494 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 ↔ 𝜓) |
6 | 5 | ralbii 3118 | . . . 4 ⊢ (∀𝑦 ∈ 𝑉 𝑅 ∈ 𝑉 ↔ ∀𝑦 ∈ 𝑉 𝜓) |
7 | 6 | ralbii 3118 | . . 3 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑅 ∈ 𝑉 ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝜓) |
8 | df-ral 3055 | . . . 4 ⊢ (∀𝑦 ∈ 𝑉 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝑉 → 𝜓)) | |
9 | 8 | ralbii 3118 | . . 3 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝜓 ↔ ∀𝑥 ∈ 𝑉 ∀𝑦(𝑦 ∈ 𝑉 → 𝜓)) |
10 | df-ral 3055 | . . 3 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦(𝑦 ∈ 𝑉 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑉 → 𝜓))) | |
11 | 7, 9, 10 | 3bitri 286 | . 2 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑅 ∈ 𝑉 ↔ ∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑉 → 𝜓))) |
12 | vex 3343 | . . . . . 6 ⊢ 𝑥 ∈ V | |
13 | cllem0.x | . . . . . 6 ⊢ (𝑧 = 𝑥 → (𝜑 ↔ 𝜒)) | |
14 | 12, 13, 4 | elab2 3494 | . . . . 5 ⊢ (𝑥 ∈ 𝑉 ↔ 𝜒) |
15 | vex 3343 | . . . . . 6 ⊢ 𝑦 ∈ V | |
16 | cllem0.y | . . . . . 6 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜃)) | |
17 | 15, 16, 4 | elab2 3494 | . . . . 5 ⊢ (𝑦 ∈ 𝑉 ↔ 𝜃) |
18 | cllem0.closed | . . . . 5 ⊢ ((𝜒 ∧ 𝜃) → 𝜓) | |
19 | 14, 17, 18 | syl2anb 497 | . . . 4 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝜓) |
20 | 19 | ex 449 | . . 3 ⊢ (𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑉 → 𝜓)) |
21 | 20 | alrimiv 2004 | . 2 ⊢ (𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑉 → 𝜓)) |
22 | 11, 21 | mpgbir 1875 | 1 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑅 ∈ 𝑉 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∀wal 1630 = wceq 1632 ∈ wcel 2139 {cab 2746 ∀wral 3050 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-v 3342 |
This theorem is referenced by: superficl 38374 superuncl 38375 ssficl 38376 ssuncl 38377 ssdifcl 38378 sssymdifcl 38379 trficl 38463 |
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