Theorem membpartlem19 39009

Description: Together with disjlem19 38999, this is former prtlem19 39077. (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 38968	. . . 4 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
2		n0el2 38467	. . . . . . . 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 2820	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → (𝑢 ∈ dom (◡ E ↾ 𝐴) ↔ 𝑢 ∈ 𝐴))
6		eldisjlem19 39008	. . . . . . . 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 1541   ∈ wcel 2113  ∅c0 4283   E cep 5521  ^◡ccnv 5621  dom cdm 5622   ↾ cres 5624  [cec 8631   ∼ ccoels 38323   ElDisj weldisj 38358   MembPart wmembpart 38363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-id 5517  df-eprel 5522  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-ec 8635  df-qs 8639  df-coss 38613  df-coels 38614  df-cnvrefrel 38719  df-dmqs 38835  df-disjALTV 38903  df-eldisj 38905  df-part 38964  df-membpart 38966

This theorem is referenced by: (None)