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Theorem xpnum 8976
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 8970 . 2 (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐴)
2 isnum2 8970 . 2 (𝐵 ∈ dom card ↔ ∃𝑦 ∈ On 𝑦𝐵)
3 reeanv 3254 . . 3 (∃𝑥 ∈ On ∃𝑦 ∈ On (𝑥𝐴𝑦𝐵) ↔ (∃𝑥 ∈ On 𝑥𝐴 ∧ ∃𝑦 ∈ On 𝑦𝐵))
4 omcl 7769 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ·𝑜 𝑦) ∈ On)
54adantr 466 . . . . . 6 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 ·𝑜 𝑦) ∈ On)
6 omxpen 8217 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ·𝑜 𝑦) ≈ (𝑥 × 𝑦))
7 xpen 8278 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → (𝑥 × 𝑦) ≈ (𝐴 × 𝐵))
8 entr 8160 . . . . . . 7 (((𝑥 ·𝑜 𝑦) ≈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ≈ (𝐴 × 𝐵)) → (𝑥 ·𝑜 𝑦) ≈ (𝐴 × 𝐵))
96, 7, 8syl2an 575 . . . . . 6 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 ·𝑜 𝑦) ≈ (𝐴 × 𝐵))
10 isnumi 8971 . . . . . 6 (((𝑥 ·𝑜 𝑦) ∈ On ∧ (𝑥 ·𝑜 𝑦) ≈ (𝐴 × 𝐵)) → (𝐴 × 𝐵) ∈ dom card)
115, 9, 10syl2anc 565 . . . . 5 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥𝐴𝑦𝐵)) → (𝐴 × 𝐵) ∈ dom card)
1211ex 397 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥𝐴𝑦𝐵) → (𝐴 × 𝐵) ∈ dom card))
1312rexlimivv 3183 . . 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 577 1 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2144  wrex 3061   class class class wbr 4784   × cxp 5247  dom cdm 5249  Oncon0 5866  (class class class)co 6792   ·𝑜 comu 7710  cen 8105  cardccrd 8960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-omul 7717  df-er 7895  df-en 8109  df-dom 8110  df-card 8964
This theorem is referenced by:  iunfictbso  9136  znnen  15146  qnnen  15147  ptcmplem2  22076  finixpnum  33720  poimirlem32  33767  isnumbasgrplem2  38193
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