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| Mirrors > Home > MPE Home > Th. List > Mathboxes > membpartlem19 | Structured version Visualization version GIF version | ||
| Description: Together with disjlem19 38802, this is former prtlem19 38879. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 21-Oct-2021.) |
| Ref | Expression |
|---|---|
| membpartlem19 | ⊢ (𝐵 ∈ 𝑉 → ( MembPart 𝐴 → ((𝑢 ∈ 𝐴 ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfmembpart2 38771 | . . . 4 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) | |
| 2 | n0el2 38334 | . . . . . . . 8 ⊢ (¬ ∅ ∈ 𝐴 ↔ dom (◡ E ↾ 𝐴) = 𝐴) | |
| 3 | 2 | biimpi 216 | . . . . . . 7 ⊢ (¬ ∅ ∈ 𝐴 → dom (◡ E ↾ 𝐴) = 𝐴) |
| 4 | 3 | ad2antll 729 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → dom (◡ E ↾ 𝐴) = 𝐴) |
| 5 | 4 | eleq2d 2827 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → (𝑢 ∈ dom (◡ E ↾ 𝐴) ↔ 𝑢 ∈ 𝐴)) |
| 6 | eldisjlem19 38811 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴))) | |
| 7 | 6 | adantrd 491 | . . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴))) |
| 8 | 7 | imp 406 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴)) |
| 9 | 8 | expd 415 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → (𝑢 ∈ dom (◡ E ↾ 𝐴) → (𝐵 ∈ 𝑢 → 𝑢 = [𝐵] ∼ 𝐴))) |
| 10 | 5, 9 | sylbird 260 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → (𝑢 ∈ 𝐴 → (𝐵 ∈ 𝑢 → 𝑢 = [𝐵] ∼ 𝐴))) |
| 11 | 1, 10 | sylan2b 594 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ MembPart 𝐴) → (𝑢 ∈ 𝐴 → (𝐵 ∈ 𝑢 → 𝑢 = [𝐵] ∼ 𝐴))) |
| 12 | 11 | impd 410 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ MembPart 𝐴) → ((𝑢 ∈ 𝐴 ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴)) |
| 13 | 12 | ex 412 | 1 ⊢ (𝐵 ∈ 𝑉 → ( MembPart 𝐴 → ((𝑢 ∈ 𝐴 ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∅c0 4333 E cep 5583 ◡ccnv 5684 dom cdm 5685 ↾ cres 5687 [cec 8743 ∼ ccoels 38183 ElDisj weldisj 38218 MembPart wmembpart 38223 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ec 8747 df-qs 8751 df-coss 38412 df-coels 38413 df-cnvrefrel 38528 df-dmqs 38640 df-disjALTV 38706 df-eldisj 38708 df-part 38767 df-membpart 38769 |
| This theorem is referenced by: (None) |
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