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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 RP, 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 3515 | . . . . . 6 ⊢ 𝑅 ∈ V |
3 | cllem0.r | . . . . . 6 ⊢ (𝑧 = 𝑅 → (𝜑 ↔ 𝜓)) | |
4 | cllem0.v | . . . . . 6 ⊢ 𝑉 = {𝑧 ∣ 𝜑} | |
5 | 2, 3, 4 | elab2 3672 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 ↔ 𝜓) |
6 | 5 | ralbii 3167 | . . . 4 ⊢ (∀𝑦 ∈ 𝑉 𝑅 ∈ 𝑉 ↔ ∀𝑦 ∈ 𝑉 𝜓) |
7 | 6 | ralbii 3167 | . . 3 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑅 ∈ 𝑉 ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝜓) |
8 | df-ral 3145 | . . . 4 ⊢ (∀𝑦 ∈ 𝑉 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝑉 → 𝜓)) | |
9 | 8 | ralbii 3167 | . . 3 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝜓 ↔ ∀𝑥 ∈ 𝑉 ∀𝑦(𝑦 ∈ 𝑉 → 𝜓)) |
10 | df-ral 3145 | . . 3 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦(𝑦 ∈ 𝑉 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑉 → 𝜓))) | |
11 | 7, 9, 10 | 3bitri 299 | . 2 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑅 ∈ 𝑉 ↔ ∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑉 → 𝜓))) |
12 | vex 3499 | . . . . . 6 ⊢ 𝑥 ∈ V | |
13 | cllem0.x | . . . . . 6 ⊢ (𝑧 = 𝑥 → (𝜑 ↔ 𝜒)) | |
14 | 12, 13, 4 | elab2 3672 | . . . . 5 ⊢ (𝑥 ∈ 𝑉 ↔ 𝜒) |
15 | vex 3499 | . . . . . 6 ⊢ 𝑦 ∈ V | |
16 | cllem0.y | . . . . . 6 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜃)) | |
17 | 15, 16, 4 | elab2 3672 | . . . . 5 ⊢ (𝑦 ∈ 𝑉 ↔ 𝜃) |
18 | cllem0.closed | . . . . 5 ⊢ ((𝜒 ∧ 𝜃) → 𝜓) | |
19 | 14, 17, 18 | syl2anb 599 | . . . 4 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝜓) |
20 | 19 | ex 415 | . . 3 ⊢ (𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑉 → 𝜓)) |
21 | 20 | alrimiv 1928 | . 2 ⊢ (𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑉 → 𝜓)) |
22 | 11, 21 | mpgbir 1800 | 1 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑅 ∈ 𝑉 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ∈ wcel 2114 {cab 2801 ∀wral 3140 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-v 3498 |
This theorem is referenced by: superficl 39933 superuncl 39934 ssficl 39935 ssuncl 39936 ssdifcl 39937 sssymdifcl 39938 trficl 40021 |
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