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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 8518 | . . 3 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | |
2 | ctex 8518 | . . 3 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V) | |
3 | undjudom 9587 | . . 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 9602 | . . . . 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 487 | . . . . 5 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → 𝐵 ≼ ω) | |
8 | omex 9100 | . . . . 5 ⊢ ω ∈ V | |
9 | djudom2 9603 | . . . . 5 ⊢ ((𝐵 ≼ ω ∧ ω ∈ V) → (ω ⊔ 𝐵) ≼ (ω ⊔ ω)) | |
10 | 7, 8, 9 | sylancl 588 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (ω ⊔ 𝐵) ≼ (ω ⊔ ω)) |
11 | domtr 8556 | . . . 4 ⊢ (((𝐴 ⊔ 𝐵) ≼ (ω ⊔ 𝐵) ∧ (ω ⊔ 𝐵) ≼ (ω ⊔ ω)) → (𝐴 ⊔ 𝐵) ≼ (ω ⊔ ω)) | |
12 | 6, 10, 11 | syl2anc 586 | . . 3 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 ⊔ 𝐵) ≼ (ω ⊔ ω)) |
13 | 8, 8 | xpex 7470 | . . . . 5 ⊢ (ω × ω) ∈ V |
14 | xp2dju 9596 | . . . . . 6 ⊢ (2o × ω) = (ω ⊔ ω) | |
15 | ordom 7583 | . . . . . . . 8 ⊢ Ord ω | |
16 | 2onn 8260 | . . . . . . . 8 ⊢ 2o ∈ ω | |
17 | ordelss 6202 | . . . . . . . 8 ⊢ ((Ord ω ∧ 2o ∈ ω) → 2o ⊆ ω) | |
18 | 15, 16, 17 | mp2an 690 | . . . . . . 7 ⊢ 2o ⊆ ω |
19 | xpss1 5569 | . . . . . . 7 ⊢ (2o ⊆ ω → (2o × ω) ⊆ (ω × ω)) | |
20 | 18, 19 | ax-mp 5 | . . . . . 6 ⊢ (2o × ω) ⊆ (ω × ω) |
21 | 14, 20 | eqsstrri 4002 | . . . . 5 ⊢ (ω ⊔ ω) ⊆ (ω × ω) |
22 | ssdomg 8549 | . . . . 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 9435 | . . . 4 ⊢ (ω × ω) ≈ ω | |
25 | domentr 8562 | . . . 4 ⊢ (((ω ⊔ ω) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (ω ⊔ ω) ≼ ω) | |
26 | 23, 24, 25 | mp2an 690 | . . 3 ⊢ (ω ⊔ ω) ≼ ω |
27 | domtr 8556 | . . 3 ⊢ (((𝐴 ⊔ 𝐵) ≼ (ω ⊔ ω) ∧ (ω ⊔ ω) ≼ ω) → (𝐴 ⊔ 𝐵) ≼ ω) | |
28 | 12, 26, 27 | sylancl 588 | . 2 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 ⊔ 𝐵) ≼ ω) |
29 | domtr 8556 | . 2 ⊢ (((𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵) ∧ (𝐴 ⊔ 𝐵) ≼ ω) → (𝐴 ∪ 𝐵) ≼ ω) | |
30 | 4, 28, 29 | syl2anc 586 | 1 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 ∪ 𝐵) ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 Vcvv 3495 ∪ cun 3934 ⊆ wss 3936 class class class wbr 5059 × cxp 5548 Ord word 6185 ωcom 7574 2oc2o 8090 ≈ cen 8500 ≼ cdom 8501 ⊔ cdju 9321 |
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 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 |
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 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-oi 8968 df-dju 9324 df-card 9362 |
This theorem is referenced by: cctop 21608 2ndcdisj2 22059 ovolctb2 24087 uniiccdif 24173 prct 30444 pibt2 34692 |
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