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| Mirrors > Home > MPE Home > Th. List > neipeltop | Structured version Visualization version GIF version | ||
| Description: Lemma for neiptopreu 23141. (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 2829 | . . . 4 ⊢ (𝑎 = 𝐶 → (𝑎 ∈ (𝑁‘𝑝) ↔ 𝐶 ∈ (𝑁‘𝑝))) | |
| 2 | 1 | raleqbi1dv 3338 | . . 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 3695 | . 2 ⊢ (𝐶 ∈ 𝐽 ↔ (𝐶 ∈ 𝒫 𝑋 ∧ ∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝))) |
| 5 | 0ex 5307 | . . . . . . 7 ⊢ ∅ ∈ V | |
| 6 | eleq1 2829 | . . . . . . 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 3501 | . . . . . . 7 ⊢ (𝐶 ∈ (𝑁‘𝑝) → 𝐶 ∈ V) | |
| 10 | 9 | ralimi 3083 | . . . . . 6 ⊢ (∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝) → ∀𝑝 ∈ 𝐶 𝐶 ∈ V) |
| 11 | r19.3rzv 4499 | . . . . . . 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 3029 | . . . 4 ⊢ (∀𝑝 ∈ 𝐶 𝐶 ∈ (𝑁‘𝑝) → 𝐶 ∈ V) |
| 15 | elpwg 4603 | . . . 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 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 {crab 3436 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-ss 3968 df-nul 4334 df-pw 4602 |
| This theorem is referenced by: neiptopuni 23138 neiptoptop 23139 neiptopnei 23140 neiptopreu 23141 |
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