Theorem sexp3 8151

Description: Show that the triple order is set-like. (Contributed by Scott Fenton, 21-Aug-2024.)

Hypotheses

Ref	Expression
xpord3.1	⊢ 𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}
sexp3.1	⊢ (𝜑 → 𝑅 Se 𝐴)
sexp3.2	⊢ (𝜑 → 𝑆 Se 𝐵)
sexp3.3	⊢ (𝜑 → 𝑇 Se 𝐶)

Assertion

Ref	Expression
sexp3	⊢ (𝜑 → 𝑈 Se ((𝐴 × 𝐵) × 𝐶))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝑈(𝑥, 𝑦)

Proof of Theorem sexp3

Dummy variables 𝑎 𝑏 𝑐 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		el2xptp 8034	. . . 4 ⊢ (𝑝 ∈ ((𝐴 × 𝐵) × 𝐶) ↔ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 𝑝 = ⟨𝑎, 𝑏, 𝑐⟩)
2		df-3an 1105	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 ∈ 𝐶))
3		xpord3.1	. . . . . . . . . . . 12 ⊢ 𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}
4	3	xpord3pred 8150	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶) → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨𝑎, 𝑏, 𝑐⟩) = ((((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) × (Pred(𝑇, 𝐶, 𝑐) ∪ {𝑐})) ∖ {⟨𝑎, 𝑏, 𝑐⟩}))
5	4	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨𝑎, 𝑏, 𝑐⟩) = ((((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) × (Pred(𝑇, 𝐶, 𝑐) ∪ {𝑐})) ∖ {⟨𝑎, 𝑏, 𝑐⟩}))
6		simpr1 1213	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑎 ∈ 𝐴)
7		sexp3.1	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 Se 𝐴)
8	7	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑅 Se 𝐴)
9		setlikespec 6330	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
10	6, 8, 9	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
11		vsnex 5409	. . . . . . . . . . . . . . 15 ⊢ {𝑎} ∈ V
12	11	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → {𝑎} ∈ V)
13	10, 12	unexd 7755	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∈ V)
14		simpr2 1214	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑏 ∈ 𝐵)
15		sexp3.2	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑆 Se 𝐵)
16	15	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑆 Se 𝐵)
17		setlikespec 6330	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝑆 Se 𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
18	14, 16, 17	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
19		vsnex 5409	. . . . . . . . . . . . . . 15 ⊢ {𝑏} ∈ V
20	19	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → {𝑏} ∈ V)
21	18, 20	unexd 7755	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}) ∈ V)
22	13, 21	xpexd 7752	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∈ V)
23		simpr3 1215	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑐 ∈ 𝐶)
24		sexp3.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑇 Se 𝐶)
25	24	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑇 Se 𝐶)
26		setlikespec 6330	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ 𝐶 ∧ 𝑇 Se 𝐶) → Pred(𝑇, 𝐶, 𝑐) ∈ V)
27	23, 25, 26	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → Pred(𝑇, 𝐶, 𝑐) ∈ V)
28		vsnex 5409	. . . . . . . . . . . . . 14 ⊢ {𝑐} ∈ V
29	28	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → {𝑐} ∈ V)
30	27, 29	unexd 7755	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (Pred(𝑇, 𝐶, 𝑐) ∪ {𝑐}) ∈ V)
31	22, 30	xpexd 7752	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) × (Pred(𝑇, 𝐶, 𝑐) ∪ {𝑐})) ∈ V)
32	31	difexd 5305	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) × (Pred(𝑇, 𝐶, 𝑐) ∪ {𝑐})) ∖ {⟨𝑎, 𝑏, 𝑐⟩}) ∈ V)
33	5, 32	eqeltrd 2866	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨𝑎, 𝑏, 𝑐⟩) ∈ V)
34		predeq3 6310	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) = Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨𝑎, 𝑏, 𝑐⟩))
35	34	eleq1d 2851	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → (Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V ↔ Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨𝑎, 𝑏, 𝑐⟩) ∈ V))
36	33, 35	syl5ibrcom 250	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
37	2, 36	sylan2br 607	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 ∈ 𝐶)) → (𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
38	37	anassrs 473	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑐 ∈ 𝐶) → (𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
39	38	rexlimdva 3169	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (∃𝑐 ∈ 𝐶 𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
40	39	rexlimdvva 3225	. . . 4 ⊢ (𝜑 → (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
41	1, 40	biimtrid 245	. . 3 ⊢ (𝜑 → (𝑝 ∈ ((𝐴 × 𝐵) × 𝐶) → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
42	41	ralrimiv 3159	. 2 ⊢ (𝜑 → ∀𝑝 ∈ ((𝐴 × 𝐵) × 𝐶)Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V)
43		dfse3 6341	. 2 ⊢ (𝑈 Se ((𝐴 × 𝐵) × 𝐶) ↔ ∀𝑝 ∈ ((𝐴 × 𝐵) × 𝐶)Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V)
44	42, 43	sylibr 237	1 ⊢ (𝜑 → 𝑈 Se ((𝐴 × 𝐵) × 𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  ∀wral 3082  ∃wrex 3092  Vcvv 3458   ∖ cdif 3905   ∪ cun 3906  {csn 4592  ⟨cotp 4600   class class class wbr 5112  {copab 5176   Se wse 5615   × cxp 5662  Predcpred 6305  ‘cfv 6540  1^st c1st 7986  2^nd c2nd 7987

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5260  ax-nul 5272  ax-pow 5339  ax-pr 5407  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-ot 4601  df-uni 4876  df-iun 4961  df-br 5113  df-opab 5177  df-mpt 5196  df-id 5559  df-se 5618  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6306  df-iota 6496  df-fun 6542  df-fv 6548  df-1st 7988  df-2nd 7989

This theorem is used by:  xpord3inddlem  8152