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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 8983 | . . . . 5 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V) | |
2 | 1 | adantl 481 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → 𝐵 ∈ V) |
3 | simpl 482 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω) | |
4 | xpdom1g 9093 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐴 ≼ ω) → (𝐴 × 𝐵) ≼ (ω × 𝐵)) | |
5 | 2, 3, 4 | syl2anc 583 | . . 3 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 × 𝐵) ≼ (ω × 𝐵)) |
6 | omex 9666 | . . . . 5 ⊢ ω ∈ V | |
7 | 6 | xpdom2 9091 | . . . 4 ⊢ (𝐵 ≼ ω → (ω × 𝐵) ≼ (ω × ω)) |
8 | 7 | adantl 481 | . . 3 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (ω × 𝐵) ≼ (ω × ω)) |
9 | domtr 9027 | . . 3 ⊢ (((𝐴 × 𝐵) ≼ (ω × 𝐵) ∧ (ω × 𝐵) ≼ (ω × ω)) → (𝐴 × 𝐵) ≼ (ω × ω)) | |
10 | 5, 8, 9 | syl2anc 583 | . 2 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 × 𝐵) ≼ (ω × ω)) |
11 | xpomen 10038 | . 2 ⊢ (ω × ω) ≈ ω | |
12 | domentr 9033 | . 2 ⊢ (((𝐴 × 𝐵) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝐴 × 𝐵) ≼ ω) | |
13 | 10, 11, 12 | sylancl 585 | 1 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 × 𝐵) ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 Vcvv 3471 class class class wbr 5148 × cxp 5676 ωcom 7870 ≈ cen 8960 ≼ cdom 8961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9664 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-oi 9533 df-card 9962 |
This theorem is referenced by: tx1stc 23553 mpocti 32497 mpct 44574 opnvonmbllem2 46021 smflimlem6 46164 smfpimbor1lem1 46186 |
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