Theorem membpartlem19 39665

Description: Together with disjlem19 39655, this is former prtlem19 39754. (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 39624	. . . 4 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
2		n0el2 39086	. . . . . . . 8 ⊢ (¬ ∅ ∈ 𝐴 ↔ dom (◡ E ↾ 𝐴) = 𝐴)
3	2	biimpi 219	. . . . . . 7 ⊢ (¬ ∅ ∈ 𝐴 → dom (◡ E ↾ 𝐴) = 𝐴)
4	3	ad2antll 742	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → dom (◡ E ↾ 𝐴) = 𝐴)
5	4	eleq2d 2846	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → (𝑢 ∈ dom (◡ E ↾ 𝐴) ↔ 𝑢 ∈ 𝐴))
6		eldisjlem19 39664	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))
7	6	adantrd 497	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))
8	7	imp 412	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → ((𝑢 ∈ dom (◡ E ↾ 𝐴) ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴))
9	8	expd 421	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → (𝑢 ∈ dom (◡ E ↾ 𝐴) → (𝐵 ∈ 𝑢 → 𝑢 = [𝐵] ∼ 𝐴)))
10	5, 9	sylbird 263	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → (𝑢 ∈ 𝐴 → (𝐵 ∈ 𝑢 → 𝑢 = [𝐵] ∼ 𝐴)))
11	1, 10	sylan2b 606	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ MembPart 𝐴) → (𝑢 ∈ 𝐴 → (𝐵 ∈ 𝑢 → 𝑢 = [𝐵] ∼ 𝐴)))
12	11	impd 416	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ MembPart 𝐴) → ((𝑢 ∈ 𝐴 ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴))
13	12	ex 418	1 ⊢ (𝐵 ∈ 𝑉 → ( MembPart 𝐴 → ((𝑢 ∈ 𝐴 ∧ 𝐵 ∈ 𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∅c0 4279   E cep 5554  ^◡ccnv 5654  dom cdm 5655   ↾ cres 5657  [cec 8697   ∼ ccoels 38935   ElDisj weldisj 38972   MembPart wmembpart 38977

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5550  df-eprel 5555  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8701  df-qs 8705  df-coss 39252  df-coels 39253  df-cnvrefrel 39358  df-dmqs 39474  df-disjALTV 39541  df-eldisj 39543  df-part 39620  df-membpart 39622

This theorem is used by: (None)