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Theorem tskxpss 10812
Description: A Cartesian product of two parts of a Tarski class is a part of the class. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Jun-2013.)
Assertion
Ref Expression
tskxpss ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ⊆ 𝑇)

Proof of Theorem tskxpss
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp2 5709 . . . . 5 (𝑧 ∈ (𝑇 × 𝑇) ↔ ∃𝑥𝑇𝑦𝑇 𝑧 = ⟨𝑥, 𝑦⟩)
2 tskop 10811 . . . . . . . 8 ((𝑇 ∈ Tarski ∧ 𝑥𝑇𝑦𝑇) → ⟨𝑥, 𝑦⟩ ∈ 𝑇)
3 eleq1a 2836 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑇 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑇))
42, 3syl 17 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑥𝑇𝑦𝑇) → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑇))
543expib 1123 . . . . . 6 (𝑇 ∈ Tarski → ((𝑥𝑇𝑦𝑇) → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑇)))
65rexlimdvv 3212 . . . . 5 (𝑇 ∈ Tarski → (∃𝑥𝑇𝑦𝑇 𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑇))
71, 6biimtrid 242 . . . 4 (𝑇 ∈ Tarski → (𝑧 ∈ (𝑇 × 𝑇) → 𝑧𝑇))
87ssrdv 3989 . . 3 (𝑇 ∈ Tarski → (𝑇 × 𝑇) ⊆ 𝑇)
9 xpss12 5700 . . 3 ((𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ⊆ (𝑇 × 𝑇))
10 sstr 3992 . . . 4 (((𝐴 × 𝐵) ⊆ (𝑇 × 𝑇) ∧ (𝑇 × 𝑇) ⊆ 𝑇) → (𝐴 × 𝐵) ⊆ 𝑇)
1110expcom 413 . . 3 ((𝑇 × 𝑇) ⊆ 𝑇 → ((𝐴 × 𝐵) ⊆ (𝑇 × 𝑇) → (𝐴 × 𝐵) ⊆ 𝑇))
128, 9, 11syl2im 40 . 2 (𝑇 ∈ Tarski → ((𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ⊆ 𝑇))
13123impib 1117 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ⊆ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wrex 3070  wss 3951  cop 4632   × cxp 5683  Tarskictsk 10788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-r1 9804  df-tsk 10789
This theorem is referenced by:  tskcard  10821
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