Theorem suceldisj 39495

Description: Disjointness of successor enforces element-carrier separation: If 𝐵 is the successor of 𝐴 and 𝐵 is element-disjoint as a family, then no element of 𝐴 can itself be a member of 𝐴 (equivalently, every 𝑥 ∈ 𝐴 has empty intersection with the carrier 𝐴). Provides a clean bridge between "disjoint family at the next grade" and "no block contains a block of the same family" at the previous grade: MembPart alone does not enforce this, see dfmembpart2 39550 (it gives disjoint blocks and excludes the empty block, but does not prevent 𝑢 ∈ 𝑚 from also being a member of the carrier 𝑚). This lemma is used to justify when grade-stability (via successor-shift) supplies the extra separation axioms needed in roof/root-style carrier reasoning. (Contributed by Peter Mazsa, 18-Feb-2026.)

Assertion

Ref	Expression
suceldisj	⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → ∀𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)

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

Proof of Theorem suceldisj

Step	Hyp	Ref	Expression
1		elirr 9560	. . . . . . 7 ⊢ ¬ 𝐴 ∈ 𝐴
2		eleq1 2850	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴))
3	1, 2	mtbiri 330	. . . . . 6 ⊢ (𝑥 = 𝐴 → ¬ 𝑥 ∈ 𝐴)
4	3	con2i 140	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑥 = 𝐴)
5	4	adantl 486	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 = 𝐴)
6		sssucid 6443	. . . . . . . . . 10 ⊢ 𝐴 ⊆ suc 𝐴
7		sseq2 3962	. . . . . . . . . 10 ⊢ (suc 𝐴 = 𝐵 → (𝐴 ⊆ suc 𝐴 ↔ 𝐴 ⊆ 𝐵))
8	6, 7	mpbii 236	. . . . . . . . 9 ⊢ (suc 𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
9	8	3ad2ant3 1152	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → 𝐴 ⊆ 𝐵)
10	9	sseld 3935	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
11		sucidg 6444	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)
12	11	3ad2ant1 1150	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → 𝐴 ∈ suc 𝐴)
13		eleq2 2851	. . . . . . . . 9 ⊢ (suc 𝐴 = 𝐵 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ 𝐵))
14	13	3ad2ant3 1152	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ 𝐵))
15	12, 14	mpbid 235	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → 𝐴 ∈ 𝐵)
16	10, 15	jctird 535	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵)))
17		eldisjim3 39492	. . . . . . 7 ⊢ ( ElDisj 𝐵 → ((𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∩ 𝐴) ≠ ∅ → 𝑥 = 𝐴)))
18	17	3ad2ant2 1151	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → ((𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∩ 𝐴) ≠ ∅ → 𝑥 = 𝐴)))
19	16, 18	syld 48	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → (𝑥 ∈ 𝐴 → ((𝑥 ∩ 𝐴) ≠ ∅ → 𝑥 = 𝐴)))
20	19	imp 411	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∩ 𝐴) ≠ ∅ → 𝑥 = 𝐴))
21	5, 20	mtod 201	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) ∧ 𝑥 ∈ 𝐴) → ¬ (𝑥 ∩ 𝐴) ≠ ∅)
22		nne 2961	. . 3 ⊢ (¬ (𝑥 ∩ 𝐴) ≠ ∅ ↔ (𝑥 ∩ 𝐴) = ∅)
23	21, 22	sylib 221	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∩ 𝐴) = ∅)
24	23	ralrimiva 3156	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → ∀𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  suc csuc 6362   ElDisj weldisj 38898

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403  ax-reg 9552

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-id 5555  df-eprel 5560  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-suc 6366  df-ec 8694  df-coss 39178  df-cnvrefrel 39284  df-disjALTV 39467  df-eldisj 39469

This theorem is used by: (None)