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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 8523 | . . . . 5 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V) | |
2 | 1 | adantl 484 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → 𝐵 ∈ V) |
3 | simpl 485 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω) | |
4 | xpdom1g 8613 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐴 ≼ ω) → (𝐴 × 𝐵) ≼ (ω × 𝐵)) | |
5 | 2, 3, 4 | syl2anc 586 | . . 3 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 × 𝐵) ≼ (ω × 𝐵)) |
6 | omex 9105 | . . . . 5 ⊢ ω ∈ V | |
7 | 6 | xpdom2 8611 | . . . 4 ⊢ (𝐵 ≼ ω → (ω × 𝐵) ≼ (ω × ω)) |
8 | 7 | adantl 484 | . . 3 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (ω × 𝐵) ≼ (ω × ω)) |
9 | domtr 8561 | . . 3 ⊢ (((𝐴 × 𝐵) ≼ (ω × 𝐵) ∧ (ω × 𝐵) ≼ (ω × ω)) → (𝐴 × 𝐵) ≼ (ω × ω)) | |
10 | 5, 8, 9 | syl2anc 586 | . 2 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 × 𝐵) ≼ (ω × ω)) |
11 | xpomen 9440 | . 2 ⊢ (ω × ω) ≈ ω | |
12 | domentr 8567 | . 2 ⊢ (((𝐴 × 𝐵) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝐴 × 𝐵) ≼ ω) | |
13 | 10, 11, 12 | sylancl 588 | 1 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 × 𝐵) ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 Vcvv 3494 class class class wbr 5065 × cxp 5552 ωcom 7579 ≈ cen 8505 ≼ cdom 8506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-oi 8973 df-card 9367 |
This theorem is referenced by: tx1stc 22257 mpocti 30450 mpct 41462 opnvonmbllem2 42914 smflimlem6 43051 smfpimbor1lem1 43072 |
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