Theorem xpnum 9911

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.

Step	Hyp	Ref	Expression
1		isnum2 9905	. 2 ⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)
2		isnum2 9905	. 2 ⊢ (𝐵 ∈ dom card ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝐵)
3		reeanv 3210	. . 3 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ On (𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ↔ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 ∧ ∃𝑦 ∈ On 𝑦 ≈ 𝐵))
4		omcl 8503	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ·o 𝑦) ∈ On)
5		omxpen 9048	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ·o 𝑦) ≈ (𝑥 × 𝑦))
6		xpen 9110	. . . . . . 7 ⊢ ((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) → (𝑥 × 𝑦) ≈ (𝐴 × 𝐵))
7		entr 8980	. . . . . . 7 ⊢ (((𝑥 ·o 𝑦) ≈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ≈ (𝐴 × 𝐵)) → (𝑥 ·o 𝑦) ≈ (𝐴 × 𝐵))
8	5, 6, 7	syl2an 596	. . . . . 6 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵)) → (𝑥 ·o 𝑦) ≈ (𝐴 × 𝐵))
9		isnumi 9906	. . . . . 6 ⊢ (((𝑥 ·o 𝑦) ∈ On ∧ (𝑥 ·o 𝑦) ≈ (𝐴 × 𝐵)) → (𝐴 × 𝐵) ∈ dom card)
10	4, 8, 9	syl2an2r 685	. . . . 5 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵)) → (𝐴 × 𝐵) ∈ dom card)
11	10	ex 412	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) → (𝐴 × 𝐵) ∈ dom card))
12	11	rexlimivv 3180	. . 3 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ On (𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) → (𝐴 × 𝐵) ∈ dom card)
13	3, 12	sylbir 235	. 2 ⊢ ((∃𝑥 ∈ On 𝑥 ≈ 𝐴 ∧ ∃𝑦 ∈ On 𝑦 ≈ 𝐵) → (𝐴 × 𝐵) ∈ dom card)
14	1, 2, 13	syl2anb 598	1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ∈ dom card)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  ∃wrex 3054   class class class wbr 5110   × cxp 5639  dom cdm 5641  Oncon0 6335  (class class class)co 7390   ·_o comu 8435   ≈ cen 8918  cardccrd 9895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-oadd 8441  df-omul 8442  df-er 8674  df-en 8922  df-dom 8923  df-card 9899

This theorem is referenced by:  iunfictbso  10074  znnen  16187  qnnen  16188  ptcmplem2  23947  finixpnum  37606  poimirlem32  37653  isnumbasgrplem2  43100