![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > neipeltop | Structured version Visualization version GIF version |
Description: Lemma for neiptopreu 23156. (Contributed by Thierry Arnoux, 6-Jan-2018.) |
Ref | Expression |
---|---|
neiptop.o | ⊢ 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)} |
Ref | Expression |
---|---|
neipeltop | ⊢ (𝐶 ∈ 𝐽 ↔ (𝐶 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2826 | . . . 4 ⊢ (𝑎 = 𝐶 → (𝑎 ∈ (𝑁‘𝑝) ↔ 𝐶 ∈ (𝑁‘𝑝))) | |
2 | 1 | raleqbi1dv 3335 | . . 3 ⊢ (𝑎 = 𝐶 → (∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝) ↔ ∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝))) |
3 | neiptop.o | . . 3 ⊢ 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)} | |
4 | 2, 3 | elrab2 3697 | . 2 ⊢ (𝐶 ∈ 𝐽 ↔ (𝐶 ∈ 𝒫 𝑋 ∧ ∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝))) |
5 | 0ex 5312 | . . . . . . 7 ⊢ ∅ ∈ V | |
6 | eleq1 2826 | . . . . . . 7 ⊢ (𝐶 = ∅ → (𝐶 ∈ V ↔ ∅ ∈ V)) | |
7 | 5, 6 | mpbiri 258 | . . . . . 6 ⊢ (𝐶 = ∅ → 𝐶 ∈ V) |
8 | 7 | adantl 481 | . . . . 5 ⊢ ((∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝) ∧ 𝐶 = ∅) → 𝐶 ∈ V) |
9 | elex 3498 | . . . . . . 7 ⊢ (𝐶 ∈ (𝑁‘𝑝) → 𝐶 ∈ V) | |
10 | 9 | ralimi 3080 | . . . . . 6 ⊢ (∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝) → ∀𝑝 ∈ 𝐶 𝐶 ∈ V) |
11 | r19.3rzv 4504 | . . . . . . 7 ⊢ (𝐶 ≠ ∅ → (𝐶 ∈ V ↔ ∀𝑝 ∈ 𝐶 𝐶 ∈ V)) | |
12 | 11 | biimparc 479 | . . . . . 6 ⊢ ((∀𝑝 ∈ 𝐶 𝐶 ∈ V ∧ 𝐶 ≠ ∅) → 𝐶 ∈ V) |
13 | 10, 12 | sylan 580 | . . . . 5 ⊢ ((∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝) ∧ 𝐶 ≠ ∅) → 𝐶 ∈ V) |
14 | 8, 13 | pm2.61dane 3026 | . . . 4 ⊢ (∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝) → 𝐶 ∈ V) |
15 | elpwg 4607 | . . . 4 ⊢ (𝐶 ∈ V → (𝐶 ∈ 𝒫 𝑋 ↔ 𝐶 ⊆ 𝑋)) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝) → (𝐶 ∈ 𝒫 𝑋 ↔ 𝐶 ⊆ 𝑋)) |
17 | 16 | pm5.32ri 575 | . 2 ⊢ ((𝐶 ∈ 𝒫 𝑋 ∧ ∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝)) ↔ (𝐶 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝))) |
18 | 4, 17 | bitri 275 | 1 ⊢ (𝐶 ∈ 𝐽 ↔ (𝐶 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 {crab 3432 Vcvv 3477 ⊆ wss 3962 ∅c0 4338 𝒫 cpw 4604 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-12 2174 ax-ext 2705 ax-nul 5311 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-ss 3979 df-nul 4339 df-pw 4606 |
This theorem is referenced by: neiptopuni 23153 neiptoptop 23154 neiptopnei 23155 neiptopreu 23156 |
Copyright terms: Public domain | W3C validator |