Theorem suceldisj 39392

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 39447 (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 9562	. . . . . . 7 ⊢ ¬ 𝐴 ∈ 𝐴
2		eleq1 2857	. . . . . . 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 6444	. . . . . . . . . 10 ⊢ 𝐴 ⊆ suc 𝐴
7		sseq2 3969	. . . . . . . . . 10 ⊢ (suc 𝐴 = 𝐵 → (𝐴 ⊆ suc 𝐴 ↔ 𝐴 ⊆ 𝐵))
8	6, 7	mpbii 236	. . . . . . . . 9 ⊢ (suc 𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
9	8	3ad2ant3 1151	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → 𝐴 ⊆ 𝐵)
10	9	sseld 3942	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
11		sucidg 6445	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)
12	11	3ad2ant1 1149	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → 𝐴 ∈ suc 𝐴)
13		eleq2 2858	. . . . . . . . 9 ⊢ (suc 𝐴 = 𝐵 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ 𝐵))
14	13	3ad2ant3 1151	. . . . . . . 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 39389	. . . . . . 7 ⊢ ( ElDisj 𝐵 → ((𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∩ 𝐴) ≠ ∅ → 𝑥 = 𝐴)))
18	17	3ad2ant2 1150	. . . . . 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 2968	. . 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 3163	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → ∀𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  suc csuc 6363   ElDisj weldisj 38795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-pr 5405  ax-reg 9554

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3375  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-br 5112  df-opab 5176  df-id 5557  df-eprel 5562  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-suc 6367  df-ec 8696  df-coss 39075  df-cnvrefrel 39181  df-disjALTV 39364  df-eldisj 39366

This theorem is referenced by: (None)