Theorem setindregs 35371

Description: Set (epsilon) induction. This version of setind 9688 replaces zfregs 9673 with axregszf 35370. (Contributed by BTernaryTau, 30-Dec-2025.)

Assertion

Ref	Expression
setindregs	⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V)

Distinct variable group:   𝑥,𝐴

Proof of Theorem setindregs

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssindif0 4408	. . . . . . 7 ⊢ (𝑦 ⊆ 𝐴 ↔ (𝑦 ∩ (V ∖ 𝐴)) = ∅)
2		sseq1 3952	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
3		eleq1w 2835	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
4	2, 3	imbi12d 346	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) ↔ (𝑦 ⊆ 𝐴 → 𝑦 ∈ 𝐴)))
5	4	spvv 1998	. . . . . . 7 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → (𝑦 ⊆ 𝐴 → 𝑦 ∈ 𝐴))
6	1, 5	biimtrrid 245	. . . . . 6 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → ((𝑦 ∩ (V ∖ 𝐴)) = ∅ → 𝑦 ∈ 𝐴))
7		eldifn 4076	. . . . . 6 ⊢ (𝑦 ∈ (V ∖ 𝐴) → ¬ 𝑦 ∈ 𝐴)
8	6, 7	nsyli 157	. . . . 5 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → (𝑦 ∈ (V ∖ 𝐴) → ¬ (𝑦 ∩ (V ∖ 𝐴)) = ∅))
9	8	imp 409	. . . 4 ⊢ ((∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (V ∖ 𝐴)) → ¬ (𝑦 ∩ (V ∖ 𝐴)) = ∅)
10	9	nrexdv 3147	. . 3 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → ¬ ∃𝑦 ∈ (V ∖ 𝐴)(𝑦 ∩ (V ∖ 𝐴)) = ∅)
11		axregszf 35370	. . . 4 ⊢ ((V ∖ 𝐴) ≠ ∅ → ∃𝑦 ∈ (V ∖ 𝐴)(𝑦 ∩ (V ∖ 𝐴)) = ∅)
12	11	necon1bi 2975	. . 3 ⊢ (¬ ∃𝑦 ∈ (V ∖ 𝐴)(𝑦 ∩ (V ∖ 𝐴)) = ∅ → (V ∖ 𝐴) = ∅)
13	10, 12	syl 17	. 2 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → (V ∖ 𝐴) = ∅)
14		vdif0 4413	. 2 ⊢ (𝐴 = V ↔ (V ∖ 𝐴) = ∅)
15	13, 14	sylibr 236	1 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1548   = wceq 1550   ∈ wcel 2132  ∃wrex 3076  Vcvv 3444   ∖ cdif 3892   ∩ cin 3894   ⊆ wss 3895  ∅c0 4276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724  ax-regs 35367

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1553  df-fal 1563  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-ne 2948  df-ral 3067  df-rex 3077  df-v 3446  df-dif 3898  df-in 3902  df-ss 3912  df-nul 4277

This theorem is referenced by:  setinds2regs  35372  unir1regs  35376