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Mirrors > Home > MPE Home > Th. List > unctb | Structured version Visualization version GIF version |
Description: The union of two countable sets is countable. (Contributed by FL, 25-Aug-2006.) (Proof shortened by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
unctb | ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 ∪ 𝐵) ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ctex 8909 | . . 3 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | |
2 | ctex 8909 | . . 3 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V) | |
3 | undjudom 10111 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵)) | |
4 | 1, 2, 3 | syl2an 597 | . 2 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵)) |
5 | djudom1 10126 | . . . . 5 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ∈ V) → (𝐴 ⊔ 𝐵) ≼ (ω ⊔ 𝐵)) | |
6 | 2, 5 | sylan2 594 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 ⊔ 𝐵) ≼ (ω ⊔ 𝐵)) |
7 | simpr 486 | . . . . 5 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → 𝐵 ≼ ω) | |
8 | omex 9587 | . . . . 5 ⊢ ω ∈ V | |
9 | djudom2 10127 | . . . . 5 ⊢ ((𝐵 ≼ ω ∧ ω ∈ V) → (ω ⊔ 𝐵) ≼ (ω ⊔ ω)) | |
10 | 7, 8, 9 | sylancl 587 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (ω ⊔ 𝐵) ≼ (ω ⊔ ω)) |
11 | domtr 8953 | . . . 4 ⊢ (((𝐴 ⊔ 𝐵) ≼ (ω ⊔ 𝐵) ∧ (ω ⊔ 𝐵) ≼ (ω ⊔ ω)) → (𝐴 ⊔ 𝐵) ≼ (ω ⊔ ω)) | |
12 | 6, 10, 11 | syl2anc 585 | . . 3 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 ⊔ 𝐵) ≼ (ω ⊔ ω)) |
13 | 8, 8 | xpex 7691 | . . . . 5 ⊢ (ω × ω) ∈ V |
14 | xp2dju 10120 | . . . . . 6 ⊢ (2o × ω) = (ω ⊔ ω) | |
15 | ordom 7816 | . . . . . . . 8 ⊢ Ord ω | |
16 | 2onn 8592 | . . . . . . . 8 ⊢ 2o ∈ ω | |
17 | ordelss 6337 | . . . . . . . 8 ⊢ ((Ord ω ∧ 2o ∈ ω) → 2o ⊆ ω) | |
18 | 15, 16, 17 | mp2an 691 | . . . . . . 7 ⊢ 2o ⊆ ω |
19 | xpss1 5656 | . . . . . . 7 ⊢ (2o ⊆ ω → (2o × ω) ⊆ (ω × ω)) | |
20 | 18, 19 | ax-mp 5 | . . . . . 6 ⊢ (2o × ω) ⊆ (ω × ω) |
21 | 14, 20 | eqsstrri 3983 | . . . . 5 ⊢ (ω ⊔ ω) ⊆ (ω × ω) |
22 | ssdomg 8946 | . . . . 5 ⊢ ((ω × ω) ∈ V → ((ω ⊔ ω) ⊆ (ω × ω) → (ω ⊔ ω) ≼ (ω × ω))) | |
23 | 13, 21, 22 | mp2 9 | . . . 4 ⊢ (ω ⊔ ω) ≼ (ω × ω) |
24 | xpomen 9959 | . . . 4 ⊢ (ω × ω) ≈ ω | |
25 | domentr 8959 | . . . 4 ⊢ (((ω ⊔ ω) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (ω ⊔ ω) ≼ ω) | |
26 | 23, 24, 25 | mp2an 691 | . . 3 ⊢ (ω ⊔ ω) ≼ ω |
27 | domtr 8953 | . . 3 ⊢ (((𝐴 ⊔ 𝐵) ≼ (ω ⊔ ω) ∧ (ω ⊔ ω) ≼ ω) → (𝐴 ⊔ 𝐵) ≼ ω) | |
28 | 12, 26, 27 | sylancl 587 | . 2 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 ⊔ 𝐵) ≼ ω) |
29 | domtr 8953 | . 2 ⊢ (((𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵) ∧ (𝐴 ⊔ 𝐵) ≼ ω) → (𝐴 ∪ 𝐵) ≼ ω) | |
30 | 4, 28, 29 | syl2anc 585 | 1 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 ∪ 𝐵) ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 Vcvv 3447 ∪ cun 3912 ⊆ wss 3914 class class class wbr 5109 × cxp 5635 Ord word 6320 ωcom 7806 2oc2o 8410 ≈ cen 8886 ≼ cdom 8887 ⊔ cdju 9842 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-oi 9454 df-dju 9845 df-card 9883 |
This theorem is referenced by: cctop 22379 2ndcdisj2 22831 ovolctb2 24879 uniiccdif 24965 prct 31685 pibt2 35938 |
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