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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 8958 | . . 3 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | |
2 | ctex 8958 | . . 3 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V) | |
3 | undjudom 10161 | . . 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 10176 | . . . . 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 9637 | . . . . 5 ⊢ ω ∈ V | |
9 | djudom2 10177 | . . . . 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 9002 | . . . 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 7739 | . . . . 5 ⊢ (ω × ω) ∈ V |
14 | xp2dju 10170 | . . . . . 6 ⊢ (2o × ω) = (ω ⊔ ω) | |
15 | ordom 7864 | . . . . . . . 8 ⊢ Ord ω | |
16 | 2onn 8640 | . . . . . . . 8 ⊢ 2o ∈ ω | |
17 | ordelss 6380 | . . . . . . . 8 ⊢ ((Ord ω ∧ 2o ∈ ω) → 2o ⊆ ω) | |
18 | 15, 16, 17 | mp2an 690 | . . . . . . 7 ⊢ 2o ⊆ ω |
19 | xpss1 5695 | . . . . . . 7 ⊢ (2o ⊆ ω → (2o × ω) ⊆ (ω × ω)) | |
20 | 18, 19 | ax-mp 5 | . . . . . 6 ⊢ (2o × ω) ⊆ (ω × ω) |
21 | 14, 20 | eqsstrri 4017 | . . . . 5 ⊢ (ω ⊔ ω) ⊆ (ω × ω) |
22 | ssdomg 8995 | . . . . 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 10009 | . . . 4 ⊢ (ω × ω) ≈ ω | |
25 | domentr 9008 | . . . 4 ⊢ (((ω ⊔ ω) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (ω ⊔ ω) ≼ ω) | |
26 | 23, 24, 25 | mp2an 690 | . . 3 ⊢ (ω ⊔ ω) ≼ ω |
27 | domtr 9002 | . . 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 9002 | . 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 3474 ∪ cun 3946 ⊆ wss 3948 class class class wbr 5148 × cxp 5674 Ord word 6363 ωcom 7854 2oc2o 8459 ≈ cen 8935 ≼ cdom 8936 ⊔ cdju 9892 |
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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-oi 9504 df-dju 9895 df-card 9933 |
This theorem is referenced by: cctop 22508 2ndcdisj2 22960 ovolctb2 25008 uniiccdif 25094 prct 31934 pibt2 36293 |
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