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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 8902 | . . 3 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | |
2 | ctex 8902 | . . 3 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V) | |
3 | undjudom 10102 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵)) | |
4 | 1, 2, 3 | syl2an 596 | . 2 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵)) |
5 | djudom1 10117 | . . . . 5 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ∈ V) → (𝐴 ⊔ 𝐵) ≼ (ω ⊔ 𝐵)) | |
6 | 2, 5 | sylan2 593 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 ⊔ 𝐵) ≼ (ω ⊔ 𝐵)) |
7 | simpr 485 | . . . . 5 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → 𝐵 ≼ ω) | |
8 | omex 9578 | . . . . 5 ⊢ ω ∈ V | |
9 | djudom2 10118 | . . . . 5 ⊢ ((𝐵 ≼ ω ∧ ω ∈ V) → (ω ⊔ 𝐵) ≼ (ω ⊔ ω)) | |
10 | 7, 8, 9 | sylancl 586 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (ω ⊔ 𝐵) ≼ (ω ⊔ ω)) |
11 | domtr 8946 | . . . 4 ⊢ (((𝐴 ⊔ 𝐵) ≼ (ω ⊔ 𝐵) ∧ (ω ⊔ 𝐵) ≼ (ω ⊔ ω)) → (𝐴 ⊔ 𝐵) ≼ (ω ⊔ ω)) | |
12 | 6, 10, 11 | syl2anc 584 | . . 3 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 ⊔ 𝐵) ≼ (ω ⊔ ω)) |
13 | 8, 8 | xpex 7686 | . . . . 5 ⊢ (ω × ω) ∈ V |
14 | xp2dju 10111 | . . . . . 6 ⊢ (2o × ω) = (ω ⊔ ω) | |
15 | ordom 7811 | . . . . . . . 8 ⊢ Ord ω | |
16 | 2onn 8587 | . . . . . . . 8 ⊢ 2o ∈ ω | |
17 | ordelss 6333 | . . . . . . . 8 ⊢ ((Ord ω ∧ 2o ∈ ω) → 2o ⊆ ω) | |
18 | 15, 16, 17 | mp2an 690 | . . . . . . 7 ⊢ 2o ⊆ ω |
19 | xpss1 5652 | . . . . . . 7 ⊢ (2o ⊆ ω → (2o × ω) ⊆ (ω × ω)) | |
20 | 18, 19 | ax-mp 5 | . . . . . 6 ⊢ (2o × ω) ⊆ (ω × ω) |
21 | 14, 20 | eqsstrri 3979 | . . . . 5 ⊢ (ω ⊔ ω) ⊆ (ω × ω) |
22 | ssdomg 8939 | . . . . 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 9950 | . . . 4 ⊢ (ω × ω) ≈ ω | |
25 | domentr 8952 | . . . 4 ⊢ (((ω ⊔ ω) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (ω ⊔ ω) ≼ ω) | |
26 | 23, 24, 25 | mp2an 690 | . . 3 ⊢ (ω ⊔ ω) ≼ ω |
27 | domtr 8946 | . . 3 ⊢ (((𝐴 ⊔ 𝐵) ≼ (ω ⊔ ω) ∧ (ω ⊔ ω) ≼ ω) → (𝐴 ⊔ 𝐵) ≼ ω) | |
28 | 12, 26, 27 | sylancl 586 | . 2 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 ⊔ 𝐵) ≼ ω) |
29 | domtr 8946 | . 2 ⊢ (((𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵) ∧ (𝐴 ⊔ 𝐵) ≼ ω) → (𝐴 ∪ 𝐵) ≼ ω) | |
30 | 4, 28, 29 | syl2anc 584 | 1 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 ∪ 𝐵) ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 Vcvv 3445 ∪ cun 3908 ⊆ wss 3910 class class class wbr 5105 × cxp 5631 Ord word 6316 ωcom 7801 2oc2o 8405 ≈ cen 8879 ≼ cdom 8880 ⊔ cdju 9833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-inf2 9576 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7312 df-ov 7359 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-oi 9445 df-dju 9836 df-card 9874 |
This theorem is referenced by: cctop 22354 2ndcdisj2 22806 ovolctb2 24854 uniiccdif 24940 prct 31575 pibt2 35878 |
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