Theorem suceldisj 39200

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 39255 (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 9509	. . . . . . 7 ⊢ ¬ 𝐴 ∈ 𝐴
2		eleq1 2829	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴))
3	1, 2	mtbiri 329	. . . . . 6 ⊢ (𝑥 = 𝐴 → ¬ 𝑥 ∈ 𝐴)
4	3	con2i 139	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑥 = 𝐴)
5	4	adantl 483	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 = 𝐴)
6		sssucid 6396	. . . . . . . . . 10 ⊢ 𝐴 ⊆ suc 𝐴
7		sseq2 3943	. . . . . . . . . 10 ⊢ (suc 𝐴 = 𝐵 → (𝐴 ⊆ suc 𝐴 ↔ 𝐴 ⊆ 𝐵))
8	6, 7	mpbii 235	. . . . . . . . 9 ⊢ (suc 𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
9	8	3ad2ant3 1142	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → 𝐴 ⊆ 𝐵)
10	9	sseld 3916	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
11		sucidg 6397	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)
12	11	3ad2ant1 1140	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → 𝐴 ∈ suc 𝐴)
13		eleq2 2830	. . . . . . . . 9 ⊢ (suc 𝐴 = 𝐵 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ 𝐵))
14	13	3ad2ant3 1142	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ 𝐵))
15	12, 14	mpbid 234	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → 𝐴 ∈ 𝐵)
16	10, 15	jctird 532	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵)))
17		eldisjim3 39197	. . . . . . 7 ⊢ ( ElDisj 𝐵 → ((𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∩ 𝐴) ≠ ∅ → 𝑥 = 𝐴)))
18	17	3ad2ant2 1141	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → ((𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∩ 𝐴) ≠ ∅ → 𝑥 = 𝐴)))
19	16, 18	syld 47	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → (𝑥 ∈ 𝐴 → ((𝑥 ∩ 𝐴) ≠ ∅ → 𝑥 = 𝐴)))
20	19	imp 408	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∩ 𝐴) ≠ ∅ → 𝑥 = 𝐴))
21	5, 20	mtod 200	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) ∧ 𝑥 ∈ 𝐴) → ¬ (𝑥 ∩ 𝐴) ≠ ∅)
22		nne 2940	. . 3 ⊢ (¬ (𝑥 ∩ 𝐴) ≠ ∅ ↔ (𝑥 ∩ 𝐴) = ∅)
23	21, 22	sylib 220	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥 ∩ 𝐴) = ∅)
24	23	ralrimiva 3133	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → ∀𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121   ≠ wne 2936  ∀wral 3055   ∩ cin 3884   ⊆ wss 3885  ∅c0 4264  suc csuc 6316   ElDisj weldisj 38603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-pr 5365  ax-reg 9501

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-id 5516  df-eprel 5521  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-suc 6320  df-ec 8639  df-coss 38883  df-cnvrefrel 38989  df-disjALTV 39172  df-eldisj 39174

This theorem is referenced by: (None)