Theorem sexp3 8100

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 7984	. . . 4 ⊢ (𝑝 ∈ ((𝐴 × 𝐵) × 𝐶) ↔ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 𝑝 = ⟨𝑎, 𝑏, 𝑐⟩)
2		df-3an 1094	. . . . . . . 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 8099	. . . . . . . . . . 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 482	. . . . . . . . . 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 1201	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑎 ∈ 𝐴)
7		sexp3.1	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 Se 𝐴)
8	7	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑅 Se 𝐴)
9		setlikespec 6283	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
10	6, 8, 9	syl2anc 590	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
11		vsnex 5371	. . . . . . . . . . . . . . 15 ⊢ {𝑎} ∈ V
12	11	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → {𝑎} ∈ V)
13	10, 12	unexd 7704	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∈ V)
14		simpr2 1202	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑏 ∈ 𝐵)
15		sexp3.2	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑆 Se 𝐵)
16	15	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑆 Se 𝐵)
17		setlikespec 6283	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝑆 Se 𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
18	14, 16, 17	syl2anc 590	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
19		vsnex 5371	. . . . . . . . . . . . . . 15 ⊢ {𝑏} ∈ V
20	19	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → {𝑏} ∈ V)
21	18, 20	unexd 7704	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}) ∈ V)
22	13, 21	xpexd 7701	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∈ V)
23		simpr3 1203	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑐 ∈ 𝐶)
24		sexp3.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑇 Se 𝐶)
25	24	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑇 Se 𝐶)
26		setlikespec 6283	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ 𝐶 ∧ 𝑇 Se 𝐶) → Pred(𝑇, 𝐶, 𝑐) ∈ V)
27	23, 25, 26	syl2anc 590	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → Pred(𝑇, 𝐶, 𝑐) ∈ V)
28		vsnex 5371	. . . . . . . . . . . . . 14 ⊢ {𝑐} ∈ V
29	28	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → {𝑐} ∈ V)
30	27, 29	unexd 7704	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (Pred(𝑇, 𝐶, 𝑐) ∪ {𝑐}) ∈ V)
31	22, 30	xpexd 7701	. . . . . . . . . . 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 5266	. . . . . . . . . 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 2840	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨𝑎, 𝑏, 𝑐⟩) ∈ V)
34		predeq3 6263	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) = Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨𝑎, 𝑏, 𝑐⟩))
35	34	eleq1d 2825	. . . . . . . . 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 248	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
37	2, 36	sylan2br 601	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 ∈ 𝐶)) → (𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
38	37	anassrs 468	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑐 ∈ 𝐶) → (𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
39	38	rexlimdva 3141	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (∃𝑐 ∈ 𝐶 𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
40	39	rexlimdvva 3197	. . . 4 ⊢ (𝜑 → (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
41	1, 40	biimtrid 243	. . 3 ⊢ (𝜑 → (𝑝 ∈ ((𝐴 × 𝐵) × 𝐶) → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
42	41	ralrimiv 3131	. 2 ⊢ (𝜑 → ∀𝑝 ∈ ((𝐴 × 𝐵) × 𝐶)Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V)
43		dfse3 6294	. 2 ⊢ (𝑈 Se ((𝐴 × 𝐵) × 𝐶) ↔ ∀𝑝 ∈ ((𝐴 × 𝐵) × 𝐶)Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V)
44	42, 43	sylibr 235	1 ⊢ (𝜑 → 𝑈 Se ((𝐴 × 𝐵) × 𝐶))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054  ∃wrex 3064  Vcvv 3432   ∖ cdif 3887   ∪ cun 3888  {csn 4562  ⟨cotp 4570   class class class wbr 5079  {copab 5141   Se wse 5576   × cxp 5623  Predcpred 6258  ‘cfv 6492  1^st c1st 7936  2^nd c2nd 7937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-ot 4571  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-se 5579  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-iota 6448  df-fun 6494  df-fv 6500  df-1st 7938  df-2nd 7939

This theorem is referenced by:  xpord3inddlem  8101