Theorem sexp3 33726

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		elxpxp 33586	. . . 4 ⊢ (𝑝 ∈ ((𝐴 × 𝐵) × 𝐶) ↔ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 𝑝 = ⟨⟨𝑎, 𝑏⟩, 𝑐⟩)
2		df-3an 1087	. . . . . . . 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 33725	. . . . . . . . . . 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 481	. . . . . . . . . 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 1192	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑎 ∈ 𝐴)
7		sexp3.1	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 Se 𝐴)
8	7	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑅 Se 𝐴)
9		setlikespec 6217	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
10	6, 8, 9	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
11		snex 5349	. . . . . . . . . . . . . . 15 ⊢ {𝑎} ∈ V
12	11	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → {𝑎} ∈ V)
13	10, 12	unexd 7582	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∈ V)
14		simpr2 1193	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑏 ∈ 𝐵)
15		sexp3.2	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑆 Se 𝐵)
16	15	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑆 Se 𝐵)
17		setlikespec 6217	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝑆 Se 𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
18	14, 16, 17	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
19		snex 5349	. . . . . . . . . . . . . . 15 ⊢ {𝑏} ∈ V
20	19	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → {𝑏} ∈ V)
21	18, 20	unexd 7582	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}) ∈ V)
22	13, 21	xpexd 7579	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∈ V)
23		simpr3 1194	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑐 ∈ 𝐶)
24		sexp3.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑇 Se 𝐶)
25	24	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑇 Se 𝐶)
26		setlikespec 6217	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ 𝐶 ∧ 𝑇 Se 𝐶) → Pred(𝑇, 𝐶, 𝑐) ∈ V)
27	23, 25, 26	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → Pred(𝑇, 𝐶, 𝑐) ∈ V)
28		snex 5349	. . . . . . . . . . . . . 14 ⊢ {𝑐} ∈ V
29	28	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → {𝑐} ∈ V)
30	27, 29	unexd 7582	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (Pred(𝑇, 𝐶, 𝑐) ∪ {𝑐}) ∈ V)
31	22, 30	xpexd 7579	. . . . . . . . . . 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 5248	. . . . . . . . . 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 2839	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨⟨𝑎, 𝑏⟩, 𝑐⟩) ∈ V)
34		predeq3 6195	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨⟨𝑎, 𝑏⟩, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) = Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨⟨𝑎, 𝑏⟩, 𝑐⟩))
35	34	eleq1d 2823	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨⟨𝑎, 𝑏⟩, 𝑐⟩ → (Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V ↔ Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨⟨𝑎, 𝑏⟩, 𝑐⟩) ∈ V))
36	35	biimprd 247	. . . . . . . . . 10 ⊢ (𝑝 = ⟨⟨𝑎, 𝑏⟩, 𝑐⟩ → (Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨⟨𝑎, 𝑏⟩, 𝑐⟩) ∈ V → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
37	36	com12 32	. . . . . . . . 9 ⊢ (Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨⟨𝑎, 𝑏⟩, 𝑐⟩) ∈ V → (𝑝 = ⟨⟨𝑎, 𝑏⟩, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
38	33, 37	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑝 = ⟨⟨𝑎, 𝑏⟩, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
39	2, 38	sylan2br 594	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 ∈ 𝐶)) → (𝑝 = ⟨⟨𝑎, 𝑏⟩, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
40	39	anassrs 467	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑐 ∈ 𝐶) → (𝑝 = ⟨⟨𝑎, 𝑏⟩, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
41	40	rexlimdva 3212	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (∃𝑐 ∈ 𝐶 𝑝 = ⟨⟨𝑎, 𝑏⟩, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
42	41	rexlimdvva 3222	. . . 4 ⊢ (𝜑 → (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 𝑝 = ⟨⟨𝑎, 𝑏⟩, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
43	1, 42	syl5bi 241	. . 3 ⊢ (𝜑 → (𝑝 ∈ ((𝐴 × 𝐵) × 𝐶) → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
44	43	ralrimiv 3106	. 2 ⊢ (𝜑 → ∀𝑝 ∈ ((𝐴 × 𝐵) × 𝐶)Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V)
45		dfse3 33580	. 2 ⊢ (𝑈 Se ((𝐴 × 𝐵) × 𝐶) ↔ ∀𝑝 ∈ ((𝐴 × 𝐵) × 𝐶)Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V)
46	44, 45	sylibr 233	1 ⊢ (𝜑 → 𝑈 Se ((𝐴 × 𝐵) × 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∖ cdif 3880   ∪ cun 3881  {csn 4558  ⟨cop 4564   class class class wbr 5070  {copab 5132   Se wse 5533   × cxp 5578  Predcpred 6190  ‘cfv 6418  1^st c1st 7802  2^nd c2nd 7803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-se 5536  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-iota 6376  df-fun 6420  df-fv 6426  df-1st 7804  df-2nd 7805

This theorem is referenced by:  xpord3ind  33727