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Theorem tskxpss 9473
 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 5056 . . . . 5 (𝑧 ∈ (𝑇 × 𝑇) ↔ ∃𝑥𝑇𝑦𝑇 𝑧 = ⟨𝑥, 𝑦⟩)
2 tskop 9472 . . . . . . . 8 ((𝑇 ∈ Tarski ∧ 𝑥𝑇𝑦𝑇) → ⟨𝑥, 𝑦⟩ ∈ 𝑇)
3 eleq1a 2683 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑇 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑇))
42, 3syl 17 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑥𝑇𝑦𝑇) → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑇))
543expib 1260 . . . . . 6 (𝑇 ∈ Tarski → ((𝑥𝑇𝑦𝑇) → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑇)))
65rexlimdvv 3019 . . . . 5 (𝑇 ∈ Tarski → (∃𝑥𝑇𝑦𝑇 𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑇))
71, 6syl5bi 231 . . . 4 (𝑇 ∈ Tarski → (𝑧 ∈ (𝑇 × 𝑇) → 𝑧𝑇))
87ssrdv 3574 . . 3 (𝑇 ∈ Tarski → (𝑇 × 𝑇) ⊆ 𝑇)
9 xpss12 5148 . . 3 ((𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ⊆ (𝑇 × 𝑇))
10 sstr 3576 . . . 4 (((𝐴 × 𝐵) ⊆ (𝑇 × 𝑇) ∧ (𝑇 × 𝑇) ⊆ 𝑇) → (𝐴 × 𝐵) ⊆ 𝑇)
1110expcom 450 . . 3 ((𝑇 × 𝑇) ⊆ 𝑇 → ((𝐴 × 𝐵) ⊆ (𝑇 × 𝑇) → (𝐴 × 𝐵) ⊆ 𝑇))
128, 9, 11syl2im 39 . 2 (𝑇 ∈ Tarski → ((𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ⊆ 𝑇))
13123impib 1254 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ⊆ 𝑇)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ⊆ wss 3540  ⟨cop 4131   × cxp 5036  Tarskictsk 9449 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-r1 8510  df-tsk 9450 This theorem is referenced by:  tskcard  9482
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