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Mirrors > Home > MPE Home > Th. List > neipeltop | Structured version Visualization version GIF version |
Description: Lemma for neiptopreu 21733. (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 2898 | . . . 4 ⊢ (𝑎 = 𝐶 → (𝑎 ∈ (𝑁‘𝑝) ↔ 𝐶 ∈ (𝑁‘𝑝))) | |
2 | 1 | raleqbi1dv 3402 | . . 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 3681 | . 2 ⊢ (𝐶 ∈ 𝐽 ↔ (𝐶 ∈ 𝒫 𝑋 ∧ ∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝))) |
5 | 0ex 5202 | . . . . . . 7 ⊢ ∅ ∈ V | |
6 | eleq1 2898 | . . . . . . 7 ⊢ (𝐶 = ∅ → (𝐶 ∈ V ↔ ∅ ∈ V)) | |
7 | 5, 6 | mpbiri 260 | . . . . . 6 ⊢ (𝐶 = ∅ → 𝐶 ∈ V) |
8 | 7 | adantl 484 | . . . . 5 ⊢ ((∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝) ∧ 𝐶 = ∅) → 𝐶 ∈ V) |
9 | elex 3511 | . . . . . . 7 ⊢ (𝐶 ∈ (𝑁‘𝑝) → 𝐶 ∈ V) | |
10 | 9 | ralimi 3158 | . . . . . 6 ⊢ (∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝) → ∀𝑝 ∈ 𝐶 𝐶 ∈ V) |
11 | r19.3rzv 4442 | . . . . . . 7 ⊢ (𝐶 ≠ ∅ → (𝐶 ∈ V ↔ ∀𝑝 ∈ 𝐶 𝐶 ∈ V)) | |
12 | 11 | biimparc 482 | . . . . . 6 ⊢ ((∀𝑝 ∈ 𝐶 𝐶 ∈ V ∧ 𝐶 ≠ ∅) → 𝐶 ∈ V) |
13 | 10, 12 | sylan 582 | . . . . 5 ⊢ ((∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝) ∧ 𝐶 ≠ ∅) → 𝐶 ∈ V) |
14 | 8, 13 | pm2.61dane 3102 | . . . 4 ⊢ (∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝) → 𝐶 ∈ V) |
15 | elpwg 4543 | . . . 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 578 | . 2 ⊢ ((𝐶 ∈ 𝒫 𝑋 ∧ ∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝)) ↔ (𝐶 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝))) |
18 | 4, 17 | bitri 277 | 1 ⊢ (𝐶 ∈ 𝐽 ↔ (𝐶 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ∀wral 3136 {crab 3140 Vcvv 3493 ⊆ wss 3934 ∅c0 4289 𝒫 cpw 4537 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-nul 5201 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rab 3145 df-v 3495 df-dif 3937 df-in 3941 df-ss 3950 df-nul 4290 df-pw 4539 |
This theorem is referenced by: neiptopuni 21730 neiptoptop 21731 neiptopnei 21732 neiptopreu 21733 |
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