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Mirrors > Home > MPE Home > Th. List > xpct | Structured version Visualization version GIF version |
Description: The cartesian product of two countable sets is countable. (Contributed by Thierry Arnoux, 24-Sep-2017.) |
Ref | Expression |
---|---|
xpct | ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 × 𝐵) ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ctex 8753 | . . . . 5 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V) | |
2 | 1 | adantl 482 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → 𝐵 ∈ V) |
3 | simpl 483 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω) | |
4 | xpdom1g 8856 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐴 ≼ ω) → (𝐴 × 𝐵) ≼ (ω × 𝐵)) | |
5 | 2, 3, 4 | syl2anc 584 | . . 3 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 × 𝐵) ≼ (ω × 𝐵)) |
6 | omex 9401 | . . . . 5 ⊢ ω ∈ V | |
7 | 6 | xpdom2 8854 | . . . 4 ⊢ (𝐵 ≼ ω → (ω × 𝐵) ≼ (ω × ω)) |
8 | 7 | adantl 482 | . . 3 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (ω × 𝐵) ≼ (ω × ω)) |
9 | domtr 8793 | . . 3 ⊢ (((𝐴 × 𝐵) ≼ (ω × 𝐵) ∧ (ω × 𝐵) ≼ (ω × ω)) → (𝐴 × 𝐵) ≼ (ω × ω)) | |
10 | 5, 8, 9 | syl2anc 584 | . 2 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 × 𝐵) ≼ (ω × ω)) |
11 | xpomen 9771 | . 2 ⊢ (ω × ω) ≈ ω | |
12 | domentr 8799 | . 2 ⊢ (((𝐴 × 𝐵) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝐴 × 𝐵) ≼ ω) | |
13 | 10, 11, 12 | sylancl 586 | 1 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 × 𝐵) ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 Vcvv 3432 class class class wbr 5074 × cxp 5587 ωcom 7712 ≈ cen 8730 ≼ cdom 8731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-oi 9269 df-card 9697 |
This theorem is referenced by: tx1stc 22801 mpocti 31050 mpct 42741 opnvonmbllem2 44171 smflimlem6 44311 smfpimbor1lem1 44332 |
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