Theorem xpnum 9229

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 9223	. 2 ⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)
2		isnum2 9223	. 2 ⊢ (𝐵 ∈ dom card ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝐵)
3		reeanv 3327	. . 3 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ On (𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ↔ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 ∧ ∃𝑦 ∈ On 𝑦 ≈ 𝐵))
4		omcl 8015	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ·o 𝑦) ∈ On)
5		omxpen 8469	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ·o 𝑦) ≈ (𝑥 × 𝑦))
6		xpen 8530	. . . . . . 7 ⊢ ((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) → (𝑥 × 𝑦) ≈ (𝐴 × 𝐵))
7		entr 8412	. . . . . . 7 ⊢ (((𝑥 ·o 𝑦) ≈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ≈ (𝐴 × 𝐵)) → (𝑥 ·o 𝑦) ≈ (𝐴 × 𝐵))
8	5, 6, 7	syl2an 595	. . . . . 6 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵)) → (𝑥 ·o 𝑦) ≈ (𝐴 × 𝐵))
9		isnumi 9224	. . . . . 6 ⊢ (((𝑥 ·o 𝑦) ∈ On ∧ (𝑥 ·o 𝑦) ≈ (𝐴 × 𝐵)) → (𝐴 × 𝐵) ∈ dom card)
10	4, 8, 9	syl2an2r 681	. . . . 5 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵)) → (𝐴 × 𝐵) ∈ dom card)
11	10	ex 413	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) → (𝐴 × 𝐵) ∈ dom card))
12	11	rexlimivv 3254	. . 3 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ On (𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) → (𝐴 × 𝐵) ∈ dom card)
13	3, 12	sylbir 236	. 2 ⊢ ((∃𝑥 ∈ On 𝑥 ≈ 𝐴 ∧ ∃𝑦 ∈ On 𝑦 ≈ 𝐵) → (𝐴 × 𝐵) ∈ dom card)
14	1, 2, 13	syl2anb 597	1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ∈ dom card)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2080  ∃wrex 3105   class class class wbr 4964   × cxp 5444  dom cdm 5446  Oncon0 6069  (class class class)co 7019   ·_o comu 7954   ≈ cen 8357  cardccrd 9213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-rep 5084  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-reu 3111  df-rmo 3112  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-pss 3878  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-tp 4479  df-op 4481  df-uni 4748  df-int 4785  df-iun 4829  df-br 4965  df-opab 5027  df-mpt 5044  df-tr 5067  df-id 5351  df-eprel 5356  df-po 5365  df-so 5366  df-fr 5405  df-we 5407  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-pred 6026  df-ord 6072  df-on 6073  df-lim 6074  df-suc 6075  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-f1 6233  df-fo 6234  df-f1o 6235  df-fv 6236  df-ov 7022  df-oprab 7023  df-mpo 7024  df-om 7440  df-1st 7548  df-2nd 7549  df-wrecs 7801  df-recs 7863  df-rdg 7901  df-1o 7956  df-oadd 7960  df-omul 7961  df-er 8142  df-en 8361  df-dom 8362  df-card 9217

This theorem is referenced by:  iunfictbso  9389  znnen  15398  qnnen  15399  ptcmplem2  22345  finixpnum  34421  poimirlem32  34468  isnumbasgrplem2  39202