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Theorem sexp3 8090
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 ‘(1st𝑥))𝑅(1st ‘(1st𝑦)) ∨ (1st ‘(1st𝑥)) = (1st ‘(1st𝑦))) ∧ ((2nd ‘(1st𝑥))𝑆(2nd ‘(1st𝑦)) ∨ (2nd ‘(1st𝑥)) = (2nd ‘(1st𝑦))) ∧ ((2nd𝑥)𝑇(2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦))) ∧ 𝑥𝑦))}
sexp3.1 (𝜑𝑅 Se 𝐴)
sexp3.2 (𝜑𝑆 Se 𝐵)
sexp3.3 (𝜑𝑇 Se 𝐶)
Assertion
Ref Expression
sexp3 (𝜑𝑈 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.
StepHypRef Expression
1 el2xptp 7972 . . . 4 (𝑝 ∈ ((𝐴 × 𝐵) × 𝐶) ↔ ∃𝑎𝐴𝑏𝐵𝑐𝐶 𝑝 = ⟨𝑎, 𝑏, 𝑐⟩)
2 df-3an 1090 . . . . . . . 8 ((𝑎𝐴𝑏𝐵𝑐𝐶) ↔ ((𝑎𝐴𝑏𝐵) ∧ 𝑐𝐶))
3 xpord3.1 . . . . . . . . . . . 12 𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st𝑥))𝑅(1st ‘(1st𝑦)) ∨ (1st ‘(1st𝑥)) = (1st ‘(1st𝑦))) ∧ ((2nd ‘(1st𝑥))𝑆(2nd ‘(1st𝑦)) ∨ (2nd ‘(1st𝑥)) = (2nd ‘(1st𝑦))) ∧ ((2nd𝑥)𝑇(2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦))) ∧ 𝑥𝑦))}
43xpord3pred 8089 . . . . . . . . . . 11 ((𝑎𝐴𝑏𝐵𝑐𝐶) → 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⟩}))
54adantl 483 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → 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 1195 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → 𝑎𝐴)
7 sexp3.1 . . . . . . . . . . . . . . . 16 (𝜑𝑅 Se 𝐴)
87adantr 482 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → 𝑅 Se 𝐴)
9 setlikespec 6284 . . . . . . . . . . . . . . 15 ((𝑎𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
106, 8, 9syl2anc 585 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → Pred(𝑅, 𝐴, 𝑎) ∈ V)
11 vsnex 5391 . . . . . . . . . . . . . . 15 {𝑎} ∈ V
1211a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → {𝑎} ∈ V)
1310, 12unexd 7693 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → (Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) ∈ V)
14 simpr2 1196 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → 𝑏𝐵)
15 sexp3.2 . . . . . . . . . . . . . . . 16 (𝜑𝑆 Se 𝐵)
1615adantr 482 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → 𝑆 Se 𝐵)
17 setlikespec 6284 . . . . . . . . . . . . . . 15 ((𝑏𝐵𝑆 Se 𝐵) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
1814, 16, 17syl2anc 585 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → Pred(𝑆, 𝐵, 𝑏) ∈ V)
19 vsnex 5391 . . . . . . . . . . . . . . 15 {𝑏} ∈ V
2019a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → {𝑏} ∈ V)
2118, 20unexd 7693 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏}) ∈ V)
2213, 21xpexd 7690 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → ((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) ∈ V)
23 simpr3 1197 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → 𝑐𝐶)
24 sexp3.3 . . . . . . . . . . . . . . 15 (𝜑𝑇 Se 𝐶)
2524adantr 482 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → 𝑇 Se 𝐶)
26 setlikespec 6284 . . . . . . . . . . . . . 14 ((𝑐𝐶𝑇 Se 𝐶) → Pred(𝑇, 𝐶, 𝑐) ∈ V)
2723, 25, 26syl2anc 585 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → Pred(𝑇, 𝐶, 𝑐) ∈ V)
28 vsnex 5391 . . . . . . . . . . . . . 14 {𝑐} ∈ V
2928a1i 11 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → {𝑐} ∈ V)
3027, 29unexd 7693 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → (Pred(𝑇, 𝐶, 𝑐) ∪ {𝑐}) ∈ V)
3122, 30xpexd 7690 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → (((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) × (Pred(𝑇, 𝐶, 𝑐) ∪ {𝑐})) ∈ V)
3231difexd 5291 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → ((((Pred(𝑅, 𝐴, 𝑎) ∪ {𝑎}) × (Pred(𝑆, 𝐵, 𝑏) ∪ {𝑏})) × (Pred(𝑇, 𝐶, 𝑐) ∪ {𝑐})) ∖ {⟨𝑎, 𝑏, 𝑐⟩}) ∈ V)
335, 32eqeltrd 2838 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨𝑎, 𝑏, 𝑐⟩) ∈ V)
34 predeq3 6262 . . . . . . . . . 10 (𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) = Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨𝑎, 𝑏, 𝑐⟩))
3534eleq1d 2823 . . . . . . . . 9 (𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → (Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V ↔ Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨𝑎, 𝑏, 𝑐⟩) ∈ V))
3633, 35syl5ibrcom 247 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐴𝑏𝐵𝑐𝐶)) → (𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
372, 36sylan2br 596 . . . . . . 7 ((𝜑 ∧ ((𝑎𝐴𝑏𝐵) ∧ 𝑐𝐶)) → (𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
3837anassrs 469 . . . . . 6 (((𝜑 ∧ (𝑎𝐴𝑏𝐵)) ∧ 𝑐𝐶) → (𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
3938rexlimdva 3153 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → (∃𝑐𝐶 𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
4039rexlimdvva 3206 . . . 4 (𝜑 → (∃𝑎𝐴𝑏𝐵𝑐𝐶 𝑝 = ⟨𝑎, 𝑏, 𝑐⟩ → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
411, 40biimtrid 241 . . 3 (𝜑 → (𝑝 ∈ ((𝐴 × 𝐵) × 𝐶) → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V))
4241ralrimiv 3143 . 2 (𝜑 → ∀𝑝 ∈ ((𝐴 × 𝐵) × 𝐶)Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V)
43 dfse3 6295 . 2 (𝑈 Se ((𝐴 × 𝐵) × 𝐶) ↔ ∀𝑝 ∈ ((𝐴 × 𝐵) × 𝐶)Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), 𝑝) ∈ V)
4442, 43sylibr 233 1 (𝜑𝑈 Se ((𝐴 × 𝐵) × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wne 2944  wral 3065  wrex 3074  Vcvv 3448  cdif 3912  cun 3913  {csn 4591  cotp 4599   class class class wbr 5110  {copab 5172   Se wse 5591   × cxp 5636  Predcpred 6257  cfv 6501  1st c1st 7924  2nd c2nd 7925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-ot 4600  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-se 5594  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-iota 6453  df-fun 6503  df-fv 6509  df-1st 7926  df-2nd 7927
This theorem is referenced by:  xpord3inddlem  8091
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