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Theorem xpnum 9061
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 9055 . 2 (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐴)
2 isnum2 9055 . 2 (𝐵 ∈ dom card ↔ ∃𝑦 ∈ On 𝑦𝐵)
3 reeanv 3286 . . 3 (∃𝑥 ∈ On ∃𝑦 ∈ On (𝑥𝐴𝑦𝐵) ↔ (∃𝑥 ∈ On 𝑥𝐴 ∧ ∃𝑦 ∈ On 𝑦𝐵))
4 omcl 7854 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ·𝑜 𝑦) ∈ On)
54adantr 473 . . . . . 6 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 ·𝑜 𝑦) ∈ On)
6 omxpen 8302 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ·𝑜 𝑦) ≈ (𝑥 × 𝑦))
7 xpen 8363 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → (𝑥 × 𝑦) ≈ (𝐴 × 𝐵))
8 entr 8245 . . . . . . 7 (((𝑥 ·𝑜 𝑦) ≈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ≈ (𝐴 × 𝐵)) → (𝑥 ·𝑜 𝑦) ≈ (𝐴 × 𝐵))
96, 7, 8syl2an 590 . . . . . 6 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 ·𝑜 𝑦) ≈ (𝐴 × 𝐵))
10 isnumi 9056 . . . . . 6 (((𝑥 ·𝑜 𝑦) ∈ On ∧ (𝑥 ·𝑜 𝑦) ≈ (𝐴 × 𝐵)) → (𝐴 × 𝐵) ∈ dom card)
115, 9, 10syl2anc 580 . . . . 5 (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥𝐴𝑦𝐵)) → (𝐴 × 𝐵) ∈ dom card)
1211ex 402 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥𝐴𝑦𝐵) → (𝐴 × 𝐵) ∈ dom card))
1312rexlimivv 3215 . . 3 (∃𝑥 ∈ On ∃𝑦 ∈ On (𝑥𝐴𝑦𝐵) → (𝐴 × 𝐵) ∈ dom card)
143, 13sylbir 227 . 2 ((∃𝑥 ∈ On 𝑥𝐴 ∧ ∃𝑦 ∈ On 𝑦𝐵) → (𝐴 × 𝐵) ∈ dom card)
151, 2, 14syl2anb 592 1 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wcel 2157  wrex 3088   class class class wbr 4841   × cxp 5308  dom cdm 5310  Oncon0 5939  (class class class)co 6876   ·𝑜 comu 7795  cen 8190  cardccrd 9045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-int 4666  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-om 7298  df-1st 7399  df-2nd 7400  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-1o 7797  df-oadd 7801  df-omul 7802  df-er 7980  df-en 8194  df-dom 8195  df-card 9049
This theorem is referenced by:  iunfictbso  9221  znnen  15274  qnnen  15275  ptcmplem2  22182  finixpnum  33875  poimirlem32  33922  isnumbasgrplem2  38447
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