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Theorem tskxpss 9707
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 5241 . . . . 5 (𝑧 ∈ (𝑇 × 𝑇) ↔ ∃𝑥𝑇𝑦𝑇 𝑧 = ⟨𝑥, 𝑦⟩)
2 tskop 9706 . . . . . . . 8 ((𝑇 ∈ Tarski ∧ 𝑥𝑇𝑦𝑇) → ⟨𝑥, 𝑦⟩ ∈ 𝑇)
3 eleq1a 2798 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ 𝑇 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑇))
42, 3syl 17 . . . . . . 7 ((𝑇 ∈ Tarski ∧ 𝑥𝑇𝑦𝑇) → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑇))
543expib 1116 . . . . . 6 (𝑇 ∈ Tarski → ((𝑥𝑇𝑦𝑇) → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑇)))
65rexlimdvv 3139 . . . . 5 (𝑇 ∈ Tarski → (∃𝑥𝑇𝑦𝑇 𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧𝑇))
71, 6syl5bi 232 . . . 4 (𝑇 ∈ Tarski → (𝑧 ∈ (𝑇 × 𝑇) → 𝑧𝑇))
87ssrdv 3715 . . 3 (𝑇 ∈ Tarski → (𝑇 × 𝑇) ⊆ 𝑇)
9 xpss12 5233 . . 3 ((𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ⊆ (𝑇 × 𝑇))
10 sstr 3717 . . . 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 40 . 2 (𝑇 ∈ Tarski → ((𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ⊆ 𝑇))
13123impib 1108 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ⊆ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1596  wcel 2103  wrex 3015  wss 3680  cop 4291   × cxp 5216  Tarskictsk 9683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-inf2 8651
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-r1 8740  df-tsk 9684
This theorem is referenced by:  tskcard  9716
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