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Theorem ins3kss 4280
 Description: Subset law for Ins3k A. (Contributed by SF, 14-Jan-2015.)
Assertion
Ref Expression
ins3kss Ins3k A (11c ×k (V ×k V))

Proof of Theorem ins3kss
Dummy variables x y z t u w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2862 . . . . 5 y V
2 vex 2862 . . . . 5 z V
3 opkelins3kg 4252 . . . . 5 ((y V z V) → (⟪y, z Ins3k Atuw(y = {{t}} z = ⟪u, wt, u A)))
41, 2, 3mp2an 653 . . . 4 (⟪y, z Ins3k Atuw(y = {{t}} z = ⟪u, wt, u A))
5 opkeq12 4061 . . . . . . . 8 ((y = {{t}} z = ⟪u, w⟫) → ⟪y, z⟫ = ⟪{{t}}, ⟪u, w⟫⟫)
6 vex 2862 . . . . . . . . . . 11 t V
76snel1c 4140 . . . . . . . . . 10 {t} 1c
8 snelpw1 4146 . . . . . . . . . 10 ({{t}} 11c ↔ {t} 1c)
97, 8mpbir 200 . . . . . . . . 9 {{t}} 11c
10 vex 2862 . . . . . . . . . 10 u V
11 vex 2862 . . . . . . . . . 10 w V
1210, 11opkelxpk 4248 . . . . . . . . . 10 (⟪u, w (V ×k V) ↔ (u V w V))
1310, 11, 12mpbir2an 886 . . . . . . . . 9 u, w (V ×k V)
14 snex 4111 . . . . . . . . . 10 {{t}} V
15 opkex 4113 . . . . . . . . . 10 u, w V
1614, 15opkelxpk 4248 . . . . . . . . 9 (⟪{{t}}, ⟪u, w⟫⟫ (11c ×k (V ×k V)) ↔ ({{t}} 11c u, w (V ×k V)))
179, 13, 16mpbir2an 886 . . . . . . . 8 ⟪{{t}}, ⟪u, w⟫⟫ (11c ×k (V ×k V))
185, 17syl6eqel 2441 . . . . . . 7 ((y = {{t}} z = ⟪u, w⟫) → ⟪y, z (11c ×k (V ×k V)))
19183adant3 975 . . . . . 6 ((y = {{t}} z = ⟪u, wt, u A) → ⟪y, z (11c ×k (V ×k V)))
2019exlimiv 1634 . . . . 5 (w(y = {{t}} z = ⟪u, wt, u A) → ⟪y, z (11c ×k (V ×k V)))
2120exlimivv 1635 . . . 4 (tuw(y = {{t}} z = ⟪u, wt, u A) → ⟪y, z (11c ×k (V ×k V)))
224, 21sylbi 187 . . 3 (⟪y, z Ins3k A → ⟪y, z (11c ×k (V ×k V)))
2322gen2 1547 . 2 yz(⟪y, z Ins3k A → ⟪y, z (11c ×k (V ×k V)))
24 df-ins3k 4188 . . . 4 Ins3k A = {x yz(x = ⟪y, z tuw(y = {{t}} z = ⟪u, wt, u A))}
2524opkabssvvki 4209 . . 3 Ins3k A (V ×k V)
26 ssrelk 4211 . . 3 ( Ins3k A (V ×k V) → ( Ins3k A (11c ×k (V ×k V)) ↔ yz(⟪y, z Ins3k A → ⟪y, z (11c ×k (V ×k V)))))
2725, 26ax-mp 8 . 2 ( Ins3k A (11c ×k (V ×k V)) ↔ yz(⟪y, z Ins3k A → ⟪y, z (11c ×k (V ×k V))))
2823, 27mpbir 200 1 Ins3k A (11c ×k (V ×k V))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ⊆ wss 3257  {csn 3737  ⟪copk 4057  1cc1c 4134  ℘1cpw1 4135   ×k cxpk 4174   Ins3k cins3k 4177 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-pr 3742  df-opk 4058  df-1c 4136  df-pw1 4137  df-xpk 4185  df-ins3k 4188 This theorem is referenced by:  ins3kexg  4306
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