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Theorem xpnum 8729
Description: The cartesian product of numerable sets is numerable. (Contributed by Mario Carneiro, 3-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
xpnum ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ∈ dom card)

Proof of Theorem xpnum
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnum2 8723 . 2 (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐴)
2 isnum2 8723 . 2 (𝐵 ∈ dom card ↔ ∃𝑦 ∈ On 𝑦𝐵)
3 reeanv 3100 . . 3 (∃𝑥 ∈ On ∃𝑦 ∈ On (𝑥𝐴𝑦𝐵) ↔ (∃𝑥 ∈ On 𝑥𝐴 ∧ ∃𝑦 ∈ On 𝑦𝐵))
4 omcl 7568 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ·𝑜 𝑦) ∈ On)
54adantr 481 . . . . . 6 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 ·𝑜 𝑦) ∈ On)
6 omxpen 8014 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ·𝑜 𝑦) ≈ (𝑥 × 𝑦))
7 xpen 8075 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → (𝑥 × 𝑦) ≈ (𝐴 × 𝐵))
8 entr 7960 . . . . . . 7 (((𝑥 ·𝑜 𝑦) ≈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ≈ (𝐴 × 𝐵)) → (𝑥 ·𝑜 𝑦) ≈ (𝐴 × 𝐵))
96, 7, 8syl2an 494 . . . . . 6 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 ·𝑜 𝑦) ≈ (𝐴 × 𝐵))
10 isnumi 8724 . . . . . 6 (((𝑥 ·𝑜 𝑦) ∈ On ∧ (𝑥 ·𝑜 𝑦) ≈ (𝐴 × 𝐵)) → (𝐴 × 𝐵) ∈ dom card)
115, 9, 10syl2anc 692 . . . . 5 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥𝐴𝑦𝐵)) → (𝐴 × 𝐵) ∈ dom card)
1211ex 450 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥𝐴𝑦𝐵) → (𝐴 × 𝐵) ∈ dom card))
1312rexlimivv 3030 . . 3 (∃𝑥 ∈ On ∃𝑦 ∈ On (𝑥𝐴𝑦𝐵) → (𝐴 × 𝐵) ∈ dom card)
143, 13sylbir 225 . 2 ((∃𝑥 ∈ On 𝑥𝐴 ∧ ∃𝑦 ∈ On 𝑦𝐵) → (𝐴 × 𝐵) ∈ dom card)
151, 2, 14syl2anb 496 1 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wrex 2908   class class class wbr 4618   × cxp 5077  dom cdm 5079  Oncon0 5687  (class class class)co 6610   ·𝑜 comu 7510  cen 7904  cardccrd 8713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-omul 7517  df-er 7694  df-en 7908  df-dom 7909  df-card 8717
This theorem is referenced by:  iunfictbso  8889  znnen  14877  qnnen  14878  ptcmplem2  21780  finixpnum  33061  poimirlem32  33108  isnumbasgrplem2  37190
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