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Theorem tskxpss 9994
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 5432 . . . . 5 (𝑧 ∈ (𝑇 × 𝑇) ↔ ∃𝑥𝑇𝑦𝑇 𝑧 = ⟨𝑥, 𝑦⟩)
2 tskop 9993 . . . . . . . 8 ((𝑇 ∈ Tarski ∧ 𝑥𝑇𝑦𝑇) → ⟨𝑥, 𝑦⟩ ∈ 𝑇)
3 eleq1a 2861 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑇 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑇))
42, 3syl 17 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑥𝑇𝑦𝑇) → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑇))
543expib 1102 . . . . . 6 (𝑇 ∈ Tarski → ((𝑥𝑇𝑦𝑇) → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑇)))
65rexlimdvv 3238 . . . . 5 (𝑇 ∈ Tarski → (∃𝑥𝑇𝑦𝑇 𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑇))
71, 6syl5bi 234 . . . 4 (𝑇 ∈ Tarski → (𝑧 ∈ (𝑇 × 𝑇) → 𝑧𝑇))
87ssrdv 3866 . . 3 (𝑇 ∈ Tarski → (𝑇 × 𝑇) ⊆ 𝑇)
9 xpss12 5423 . . 3 ((𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ⊆ (𝑇 × 𝑇))
10 sstr 3868 . . . 4 (((𝐴 × 𝐵) ⊆ (𝑇 × 𝑇) ∧ (𝑇 × 𝑇) ⊆ 𝑇) → (𝐴 × 𝐵) ⊆ 𝑇)
1110expcom 406 . . 3 ((𝑇 × 𝑇) ⊆ 𝑇 → ((𝐴 × 𝐵) ⊆ (𝑇 × 𝑇) → (𝐴 × 𝐵) ⊆ 𝑇))
128, 9, 11syl2im 40 . 2 (𝑇 ∈ Tarski → ((𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ⊆ 𝑇))
13123impib 1096 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ⊆ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068   = wceq 1507  wcel 2050  wrex 3089  wss 3831  cop 4448   × cxp 5406  Tarskictsk 9970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281  ax-inf2 8900
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-int 4751  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-ov 6981  df-oprab 6982  df-mpo 6983  df-om 7399  df-wrecs 7752  df-recs 7814  df-rdg 7852  df-1o 7907  df-oadd 7911  df-er 8091  df-en 8309  df-dom 8310  df-sdom 8311  df-fin 8312  df-r1 8989  df-tsk 9971
This theorem is referenced by:  tskcard  10003
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