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Mirrors > Home > MPE Home > Th. List > xpnum | Structured version Visualization version GIF version |
Description: The cartesian product of numerable sets is numerable. (Contributed by Mario Carneiro, 3-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
xpnum | ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ∈ dom card) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isnum2 8654 | . 2 ⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) | |
2 | isnum2 8654 | . 2 ⊢ (𝐵 ∈ dom card ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝐵) | |
3 | reeanv 3086 | . . 3 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ On (𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ↔ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 ∧ ∃𝑦 ∈ On 𝑦 ≈ 𝐵)) | |
4 | omcl 7503 | . . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ·𝑜 𝑦) ∈ On) | |
5 | 4 | adantr 480 | . . . . . 6 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵)) → (𝑥 ·𝑜 𝑦) ∈ On) |
6 | omxpen 7947 | . . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ·𝑜 𝑦) ≈ (𝑥 × 𝑦)) | |
7 | xpen 8008 | . . . . . . 7 ⊢ ((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) → (𝑥 × 𝑦) ≈ (𝐴 × 𝐵)) | |
8 | entr 7894 | . . . . . . 7 ⊢ (((𝑥 ·𝑜 𝑦) ≈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ≈ (𝐴 × 𝐵)) → (𝑥 ·𝑜 𝑦) ≈ (𝐴 × 𝐵)) | |
9 | 6, 7, 8 | syl2an 493 | . . . . . 6 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵)) → (𝑥 ·𝑜 𝑦) ≈ (𝐴 × 𝐵)) |
10 | isnumi 8655 | . . . . . 6 ⊢ (((𝑥 ·𝑜 𝑦) ∈ On ∧ (𝑥 ·𝑜 𝑦) ≈ (𝐴 × 𝐵)) → (𝐴 × 𝐵) ∈ dom card) | |
11 | 5, 9, 10 | syl2anc 691 | . . . . 5 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵)) → (𝐴 × 𝐵) ∈ dom card) |
12 | 11 | ex 449 | . . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) → (𝐴 × 𝐵) ∈ dom card)) |
13 | 12 | rexlimivv 3018 | . . 3 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ On (𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) → (𝐴 × 𝐵) ∈ dom card) |
14 | 3, 13 | sylbir 224 | . 2 ⊢ ((∃𝑥 ∈ On 𝑥 ≈ 𝐴 ∧ ∃𝑦 ∈ On 𝑦 ≈ 𝐵) → (𝐴 × 𝐵) ∈ dom card) |
15 | 1, 2, 14 | syl2anb 495 | 1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 × cxp 5036 dom cdm 5038 Oncon0 5640 (class class class)co 6549 ·𝑜 comu 7445 ≈ cen 7838 cardccrd 8644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-en 7842 df-dom 7843 df-card 8648 |
This theorem is referenced by: iunfictbso 8820 znnen 14780 qnnen 14781 ptcmplem2 21667 finixpnum 32564 poimirlem32 32611 isnumbasgrplem2 36693 |
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