Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 8512 | . . 3 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | |
2 | ctex 8512 | . . 3 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V) | |
3 | undjudom 9581 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵)) | |
4 | 1, 2, 3 | syl2an 595 | . 2 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵)) |
5 | djudom1 9596 | . . . . 5 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ∈ V) → (𝐴 ⊔ 𝐵) ≼ (ω ⊔ 𝐵)) | |
6 | 2, 5 | sylan2 592 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 ⊔ 𝐵) ≼ (ω ⊔ 𝐵)) |
7 | simpr 485 | . . . . 5 ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → 𝐵 ≼ ω) | |
8 | omex 9094 | . . . . 5 ⊢ ω ∈ V | |
9 | djudom2 9597 | . . . . 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 8550 | . . . 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 7465 | . . . . 5 ⊢ (ω × ω) ∈ V |
14 | xp2dju 9590 | . . . . . 6 ⊢ (2o × ω) = (ω ⊔ ω) | |
15 | ordom 7578 | . . . . . . . 8 ⊢ Ord ω | |
16 | 2onn 8255 | . . . . . . . 8 ⊢ 2o ∈ ω | |
17 | ordelss 6200 | . . . . . . . 8 ⊢ ((Ord ω ∧ 2o ∈ ω) → 2o ⊆ ω) | |
18 | 15, 16, 17 | mp2an 688 | . . . . . . 7 ⊢ 2o ⊆ ω |
19 | xpss1 5567 | . . . . . . 7 ⊢ (2o ⊆ ω → (2o × ω) ⊆ (ω × ω)) | |
20 | 18, 19 | ax-mp 5 | . . . . . 6 ⊢ (2o × ω) ⊆ (ω × ω) |
21 | 14, 20 | eqsstrri 3999 | . . . . 5 ⊢ (ω ⊔ ω) ⊆ (ω × ω) |
22 | ssdomg 8543 | . . . . 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 9429 | . . . 4 ⊢ (ω × ω) ≈ ω | |
25 | domentr 8556 | . . . 4 ⊢ (((ω ⊔ ω) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (ω ⊔ ω) ≼ ω) | |
26 | 23, 24, 25 | mp2an 688 | . . 3 ⊢ (ω ⊔ ω) ≼ ω |
27 | domtr 8550 | . . 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 8550 | . 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 2105 Vcvv 3492 ∪ cun 3931 ⊆ wss 3933 class class class wbr 5057 × cxp 5546 Ord word 6183 ωcom 7569 2oc2o 8085 ≈ cen 8494 ≼ cdom 8495 ⊔ cdju 9315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-oi 8962 df-dju 9318 df-card 9356 |
This theorem is referenced by: cctop 21542 2ndcdisj2 21993 ovolctb2 24020 uniiccdif 24106 prct 30376 pibt2 34580 |
Copyright terms: Public domain | W3C validator |