Theorem tpssad 32487

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 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → 𝐴 ∈ 𝑉)
3		tpcomb 4731	. . . . 5 ⊢ {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐵}
4		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → ¬ 𝐵 ∈ V)
5	4	intnanrd 489	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → ¬ (𝐵 ∈ V ∧ 𝐵 ≠ 𝐶))
6		tpprceq3 4784	. . . . . 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 2781	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶})
9		tpssad.2	. . . . 5 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
10	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
11	8, 10	eqsstrrd 3999	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐶} ⊆ 𝐷)
12	2, 11	prssad 32477	. 2 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ V) → 𝐴 ∈ 𝐷)
13	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → 𝐴 ∈ 𝑉)
14		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → ¬ 𝐶 ∈ V)
15	14	intnanrd 489	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → ¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐵))
16		tpprceq3 4784	. . . . 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 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
19	17, 18	eqsstrrd 3999	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ⊆ 𝐷)
20	13, 19	prssad 32477	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ V) → 𝐴 ∈ 𝐷)
21	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ∈ 𝑉)
22		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐵 ∈ V)
23		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐶 ∈ V)
24	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
25		tpssg 32485	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷))
26	25	biimpar 477	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ {𝐴, 𝐵, 𝐶} ⊆ 𝐷) → (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷))
27	21, 22, 23, 24, 26	syl31anc 1374	. . 3 ⊢ ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷))
28	27	simp1d 1142	. 2 ⊢ ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ∈ 𝐷)
29	12, 20, 28	pm2.61dda 814	1 ⊢ (𝜑 → 𝐴 ∈ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ≠ wne 2931  Vcvv 3463   ⊆ wss 3931  {cpr 4608  {ctp 4610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-sn 4607  df-pr 4609  df-tp 4611

This theorem is referenced by:  tpssbd  32488  tpsscd  32489  constrlccllem  33733  constrcccllem  33734