Theorem membpartlem19 38776

Description: Together with disjlem19 38766, this is former prtlem19 38844. (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 38735	. . . 4 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
2		n0el2 38290	. . . . . . . 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 2814	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → (𝑢 ∈ dom (◡ E ↾ 𝐴) ↔ 𝑢 ∈ 𝐴))
6		eldisjlem19 38775	. . . . . . . 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 4292   E cep 5530  ^◡ccnv 5630  dom cdm 5631   ↾ cres 5633  [cec 8646   ∼ ccoels 38143   ElDisj weldisj 38178   MembPart wmembpart 38183

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 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-id 5526  df-eprel 5531  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ec 8650  df-qs 8654  df-coss 38375  df-coels 38376  df-cnvrefrel 38491  df-dmqs 38603  df-disjALTV 38670  df-eldisj 38672  df-part 38731  df-membpart 38733

This theorem is referenced by: (None)