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Theorem xpnum 9364
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 9358 . 2 (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐴)
2 isnum2 9358 . 2 (𝐵 ∈ dom card ↔ ∃𝑦 ∈ On 𝑦𝐵)
3 reeanv 3320 . . 3 (∃𝑥 ∈ On ∃𝑦 ∈ On (𝑥𝐴𝑦𝐵) ↔ (∃𝑥 ∈ On 𝑥𝐴 ∧ ∃𝑦 ∈ On 𝑦𝐵))
4 omcl 8144 . . . . . 6 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ·o 𝑦) ∈ On)
5 omxpen 8602 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ·o 𝑦) ≈ (𝑥 × 𝑦))
6 xpen 8664 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → (𝑥 × 𝑦) ≈ (𝐴 × 𝐵))
7 entr 8544 . . . . . . 7 (((𝑥 ·o 𝑦) ≈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ≈ (𝐴 × 𝐵)) → (𝑥 ·o 𝑦) ≈ (𝐴 × 𝐵))
85, 6, 7syl2an 598 . . . . . 6 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 ·o 𝑦) ≈ (𝐴 × 𝐵))
9 isnumi 9359 . . . . . 6 (((𝑥 ·o 𝑦) ∈ On ∧ (𝑥 ·o 𝑦) ≈ (𝐴 × 𝐵)) → (𝐴 × 𝐵) ∈ dom card)
104, 8, 9syl2an2r 684 . . . . 5 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥𝐴𝑦𝐵)) → (𝐴 × 𝐵) ∈ dom card)
1110ex 416 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥𝐴𝑦𝐵) → (𝐴 × 𝐵) ∈ dom card))
1211rexlimivv 3251 . . 3 (∃𝑥 ∈ On ∃𝑦 ∈ On (𝑥𝐴𝑦𝐵) → (𝐴 × 𝐵) ∈ dom card)
133, 12sylbir 238 . 2 ((∃𝑥 ∈ On 𝑥𝐴 ∧ ∃𝑦 ∈ On 𝑦𝐵) → (𝐴 × 𝐵) ∈ dom card)
141, 2, 13syl2anb 600 1 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2111  wrex 3107   class class class wbr 5030   × cxp 5517  dom cdm 5519  Oncon0 6159  (class class class)co 7135   ·o comu 8083  cen 8489  cardccrd 9348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-omul 8090  df-er 8272  df-en 8493  df-dom 8494  df-card 9352
This theorem is referenced by:  iunfictbso  9525  znnen  15557  qnnen  15558  ptcmplem2  22658  finixpnum  35042  poimirlem32  35089  isnumbasgrplem2  40046
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