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Theorem tskxpss 10183
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 5556 . . . . 5 (𝑧 ∈ (𝑇 × 𝑇) ↔ ∃𝑥𝑇𝑦𝑇 𝑧 = ⟨𝑥, 𝑦⟩)
2 tskop 10182 . . . . . . . 8 ((𝑇 ∈ Tarski ∧ 𝑥𝑇𝑦𝑇) → ⟨𝑥, 𝑦⟩ ∈ 𝑇)
3 eleq1a 2909 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑇 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑇))
42, 3syl 17 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑥𝑇𝑦𝑇) → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑇))
543expib 1119 . . . . . 6 (𝑇 ∈ Tarski → ((𝑥𝑇𝑦𝑇) → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑇)))
65rexlimdvv 3279 . . . . 5 (𝑇 ∈ Tarski → (∃𝑥𝑇𝑦𝑇 𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑇))
71, 6syl5bi 245 . . . 4 (𝑇 ∈ Tarski → (𝑧 ∈ (𝑇 × 𝑇) → 𝑧𝑇))
87ssrdv 3948 . . 3 (𝑇 ∈ Tarski → (𝑇 × 𝑇) ⊆ 𝑇)
9 xpss12 5547 . . 3 ((𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ⊆ (𝑇 × 𝑇))
10 sstr 3950 . . . 4 (((𝐴 × 𝐵) ⊆ (𝑇 × 𝑇) ∧ (𝑇 × 𝑇) ⊆ 𝑇) → (𝐴 × 𝐵) ⊆ 𝑇)
1110expcom 417 . . 3 ((𝑇 × 𝑇) ⊆ 𝑇 → ((𝐴 × 𝐵) ⊆ (𝑇 × 𝑇) → (𝐴 × 𝐵) ⊆ 𝑇))
128, 9, 11syl2im 40 . 2 (𝑇 ∈ Tarski → ((𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ⊆ 𝑇))
13123impib 1113 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ⊆ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2114  wrex 3131  wss 3908  cop 4545   × cxp 5530  Tarskictsk 10159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-r1 9181  df-tsk 10160
This theorem is referenced by:  tskcard  10192
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