Theorem tpssad 32687

Description: If an ordered triple is a subset of a class, the first element of the triple is an element of that class. (Contributed by Thierry Arnoux, 2-Nov-2025.)

Hypotheses

Ref	Expression
tpssad.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
tpssad.2	⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Assertion

Ref	Expression
tpssad	⊢ (𝜑 → 𝐴 ∈ 𝐷)

Proof of Theorem tpssad

Step	Hyp	Ref	Expression
1		tpssad.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2	1	adantr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → 𝐴 ∈ 𝑉)
3		tpcomb 4709	. . . . 5 ⊢ {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐵}
4		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → ¬ 𝐵 ∈ V)
5	4	intnanrd 493	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → ¬ (𝐵 ∈ V ∧ 𝐵 ≠ 𝐶))
6		tpprceq3 4763	. . . . . 6 ⊢ (¬ (𝐵 ∈ V ∧ 𝐵 ≠ 𝐶) → {𝐴, 𝐶, 𝐵} = {𝐴, 𝐶})
7	5, 6	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐶, 𝐵} = {𝐴, 𝐶})
8	3, 7	eqtrid 2808	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶})
9		tpssad.2	. . . . 5 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
10	9	adantr 484	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
11	8, 10	eqsstrrd 3971	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐶} ⊆ 𝐷)
12	2, 11	prssad 32677	. 2 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → 𝐴 ∈ 𝐷)
13	1	adantr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → 𝐴 ∈ 𝑉)
14		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → ¬ 𝐶 ∈ V)
15	14	intnanrd 493	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → ¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐵))
16		tpprceq3 4763	. . . . 5 ⊢ (¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
17	15, 16	syl 17	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
18	9	adantr 484	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
19	17, 18	eqsstrrd 3971	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ⊆ 𝐷)
20	13, 19	prssad 32677	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → 𝐴 ∈ 𝐷)
21	1	adantr 484	. . . 4 ⊢ ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ∈ 𝑉)
22		simprl 780	. . . 4 ⊢ ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐵 ∈ V)
23		simprr 782	. . . 4 ⊢ ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐶 ∈ V)
24	9	adantr 484	. . . 4 ⊢ ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
25		tpssg 32685	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷))
26	25	biimpar 481	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ {𝐴, 𝐵, 𝐶} ⊆ 𝐷) → (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷))
27	21, 22, 23, 24, 26	syl31anc 1391	. . 3 ⊢ ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷))
28	27	simp1d 1154	. 2 ⊢ ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ∈ 𝐷)
29	12, 20, 28	pm2.61dda 824	1 ⊢ (𝜑 → 𝐴 ∈ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  Vcvv 3453   ⊆ wss 3904  {cpr 4583  {ctp 4585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-sn 4582  df-pr 4584  df-tp 4586

This theorem is referenced by:  tpssbd  32688  tpsscd  32689  constrlccllem  34011  constrcccllem  34012