Theorem suceldisj 39941

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 39996 (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 9618	. . . . . . 7 ⊢ ¬ 𝐴 ∈ 𝐴
2		eleq1 2852	. . . . . . 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 487	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 = 𝐴)
6		sssucid 6460	. . . . . . . . . 10 ⊢ 𝐴 ⊆ suc 𝐴
7		sseq2 3939	. . . . . . . . . 10 ⊢ (suc 𝐴 = 𝐵 → (𝐴 ⊆ suc 𝐴 ↔ 𝐴 ⊆ 𝐵))
8	6, 7	mpbii 236	. . . . . . . . 9 ⊢ (suc 𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
9	8	3ad2ant3 1153	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → 𝐴 ⊆ 𝐵)
10	9	sseld 3912	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
11		sucidg 6461	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)
12	11	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → 𝐴 ∈ suc 𝐴)
13		eleq2 2853	. . . . . . . . 9 ⊢ (suc 𝐴 = 𝐵 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ 𝐵))
14	13	3ad2ant3 1153	. . . . . . . 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 536	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵)))
17		eldisjim3 39938	. . . . . . 7 ⊢ ( ElDisj 𝐵 → ((𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∩ 𝐴) ≠ ∅ → 𝑥 = 𝐴)))
18	17	3ad2ant2 1152	. . . . . 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 412	. . . 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 2963	. . 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 3158	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 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  suc csuc 6379   ElDisj weldisj 39344

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395  ax-reg 9610

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 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  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-suc 6383  df-ec 8744  df-coss 39624  df-cnvrefrel 39730  df-disjALTV 39913  df-eldisj 39915

This theorem is used by: (None)