Theorem xpnum 9875

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 9869	. 2 ⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)
2		isnum2 9869	. 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 8473	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ·o 𝑦) ∈ On)
5		omxpen 9019	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ·o 𝑦) ≈ (𝑥 × 𝑦))
6		xpen 9080	. . . . . . 7 ⊢ ((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) → (𝑥 × 𝑦) ≈ (𝐴 × 𝐵))
7		entr 8955	. . . . . . 7 ⊢ (((𝑥 ·o 𝑦) ≈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ≈ (𝐴 × 𝐵)) → (𝑥 ·o 𝑦) ≈ (𝐴 × 𝐵))
8	5, 6, 7	syl2an 597	. . . . . 6 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵)) → (𝑥 ·o 𝑦) ≈ (𝐴 × 𝐵))
9		isnumi 9870	. . . . . 6 ⊢ (((𝑥 ·o 𝑦) ∈ On ∧ (𝑥 ·o 𝑦) ≈ (𝐴 × 𝐵)) → (𝐴 × 𝐵) ∈ dom card)
10	4, 8, 9	syl2an2r 686	. . . . 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 599	1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ∈ dom card)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  ∃wrex 3062   class class class wbr 5100   × cxp 5630  dom cdm 5632  Oncon0 6325  (class class class)co 7368   ·_o comu 8405   ≈ cen 8892  cardccrd 9859

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-oadd 8411  df-omul 8412  df-er 8645  df-en 8896  df-dom 8897  df-card 9863

This theorem is referenced by:  iunfictbso  10036  znnen  16149  qnnen  16150  ptcmplem2  24009  finixpnum  37856  poimirlem32  37903  isnumbasgrplem2  43461