Theorem membpartlem19 38810

Description: Together with disjlem19 38800, this is former prtlem19 38878. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 21-Oct-2021.)

Assertion

Ref	Expression
membpartlem19	⊢ (𝐵 ∈ 𝑉 → ( MembPart 𝐴 → ((𝑢 ∈ 𝐴 ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))

Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝑉

Proof of Theorem membpartlem19

Step	Hyp	Ref	Expression
1		dfmembpart2 38769	. . . 4 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
2		n0el2 38324	. . . . . . . 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 2815	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → (𝑢 ∈ dom (◡ E ↾ 𝐴) ↔ 𝑢 ∈ 𝐴))
6		eldisjlem19 38809	. . . . . . . 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)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∅c0 4299   E cep 5540  ^◡ccnv 5640  dom cdm 5641   ↾ cres 5643  [cec 8672   ∼ ccoels 38177   ElDisj weldisj 38212   MembPart wmembpart 38217

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-id 5536  df-eprel 5541  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ec 8676  df-qs 8680  df-coss 38409  df-coels 38410  df-cnvrefrel 38525  df-dmqs 38637  df-disjALTV 38704  df-eldisj 38706  df-part 38765  df-membpart 38767

This theorem is referenced by: (None)