Theorem sexp3 8121

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 8005	. . . 4 ⊢ (𝑝 ∈ ((𝐴 × 𝐵) × 𝐶) ↔ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 𝑝 = ⟨𝑎, 𝑏, 𝑐⟩)
2		df-3an 1097	. . . . . . . 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 8120	. . . . . . . . . . 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 484	. . . . . . . . . 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 1204	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑎 ∈ 𝐴)
7		sexp3.1	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 Se 𝐴)
8	7	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑅 Se 𝐴)
9		setlikespec 6301	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
10	6, 8, 9	syl2anc 592	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
11		vsnex 5386	. . . . . . . . . . . . . . 15 ⊢ {𝑎} ∈ V
12	11	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → {𝑎} ∈ V)
13	10, 12	unexd 7726	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∈ V)
14		simpr2 1205	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑏 ∈ 𝐵)
15		sexp3.2	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑆 Se 𝐵)
16	15	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑆 Se 𝐵)
17		setlikespec 6301	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝑆 Se 𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
18	14, 16, 17	syl2anc 592	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
19		vsnex 5386	. . . . . . . . . . . . . . 15 ⊢ {𝑏} ∈ V
20	19	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → {𝑏} ∈ V)
21	18, 20	unexd 7726	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}) ∈ V)
22	13, 21	xpexd 7723	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∈ V)
23		simpr3 1206	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑐 ∈ 𝐶)
24		sexp3.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑇 Se 𝐶)
25	24	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → 𝑇 Se 𝐶)
26		setlikespec 6301	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ 𝐶 ∧ 𝑇 Se 𝐶) → Pred(𝑇, 𝐶, 𝑐) ∈ V)
27	23, 25, 26	syl2anc 592	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → Pred(𝑇, 𝐶, 𝑐) ∈ V)
28		vsnex 5386	. . . . . . . . . . . . . 14 ⊢ {𝑐} ∈ V
29	28	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → {𝑐} ∈ V)
30	27, 29	unexd 7726	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (Pred(𝑇, 𝐶, 𝑐) ∪ {𝑐}) ∈ V)
31	22, 30	xpexd 7723	. . . . . . . . . . 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 5281	. . . . . . . . . 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 2856	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨𝑎, 𝑏, 𝑐⟩) ∈ V)
34		predeq3 6281	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) = Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨𝑎, 𝑏, 𝑐⟩))
35	34	eleq1d 2841	. . . . . . . . 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 249	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
37	2, 36	sylan2br 603	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 ∈ 𝐶)) → (𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
38	37	anassrs 470	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) ∧ 𝑐 ∈ 𝐶) → (𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
39	38	rexlimdva 3157	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (∃𝑐 ∈ 𝐶 𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
40	39	rexlimdvva 3213	. . . 4 ⊢ (𝜑 → (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
41	1, 40	biimtrid 244	. . 3 ⊢ (𝜑 → (𝑝 ∈ ((𝐴 × 𝐵) × 𝐶) → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
42	41	ralrimiv 3147	. 2 ⊢ (𝜑 → ∀𝑝 ∈ ((𝐴 × 𝐵) × 𝐶)Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V)
43		dfse3 6312	. 2 ⊢ (𝑈 Se ((𝐴 × 𝐵) × 𝐶) ↔ ∀𝑝 ∈ ((𝐴 × 𝐵) × 𝐶)Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V)
44	42, 43	sylibr 236	1 ⊢ (𝜑 → 𝑈 Se ((𝐴 × 𝐵) × 𝐶))

Syntax hints:   → wi 4   ∧ wa 398   ∨ wo 856   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136   ≠ wne 2951  ∀wral 3070  ∃wrex 3080  Vcvv 3448   ∖ cdif 3896   ∪ cun 3897  {csn 4576  ⟨cotp 4584   class class class wbr 5094  {copab 5156   Se wse 5591   × cxp 5638  Predcpred 6276  ‘cfv 6510  1^st c1st 7957  2^nd c2nd 7958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-ot 4585  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-se 5594  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6277  df-iota 6466  df-fun 6512  df-fv 6518  df-1st 7959  df-2nd 7960

This theorem is referenced by:  xpord3inddlem  8122