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Mirrors > Home > MPE Home > Th. List > djufi | Structured version Visualization version GIF version |
Description: The disjoint union of two finite sets is finite. (Contributed by NM, 22-Oct-2004.) |
Ref | Expression |
---|---|
djufi | ⊢ ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴 ⊔ 𝐵) ≺ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dju 9845 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | 0elon 6375 | . . . . . 6 ⊢ ∅ ∈ On | |
3 | relsdom 8896 | . . . . . . 7 ⊢ Rel ≺ | |
4 | 3 | brrelex1i 5692 | . . . . . 6 ⊢ (𝐴 ≺ ω → 𝐴 ∈ V) |
5 | xpsnen2g 9015 | . . . . . 6 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ V) → ({∅} × 𝐴) ≈ 𝐴) | |
6 | 2, 4, 5 | sylancr 588 | . . . . 5 ⊢ (𝐴 ≺ ω → ({∅} × 𝐴) ≈ 𝐴) |
7 | sdomen1 9071 | . . . . 5 ⊢ (({∅} × 𝐴) ≈ 𝐴 → (({∅} × 𝐴) ≺ ω ↔ 𝐴 ≺ ω)) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝐴 ≺ ω → (({∅} × 𝐴) ≺ ω ↔ 𝐴 ≺ ω)) |
9 | 8 | ibir 268 | . . 3 ⊢ (𝐴 ≺ ω → ({∅} × 𝐴) ≺ ω) |
10 | 1on 8428 | . . . . . 6 ⊢ 1o ∈ On | |
11 | 3 | brrelex1i 5692 | . . . . . 6 ⊢ (𝐵 ≺ ω → 𝐵 ∈ V) |
12 | xpsnen2g 9015 | . . . . . 6 ⊢ ((1o ∈ On ∧ 𝐵 ∈ V) → ({1o} × 𝐵) ≈ 𝐵) | |
13 | 10, 11, 12 | sylancr 588 | . . . . 5 ⊢ (𝐵 ≺ ω → ({1o} × 𝐵) ≈ 𝐵) |
14 | sdomen1 9071 | . . . . 5 ⊢ (({1o} × 𝐵) ≈ 𝐵 → (({1o} × 𝐵) ≺ ω ↔ 𝐵 ≺ ω)) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝐵 ≺ ω → (({1o} × 𝐵) ≺ ω ↔ 𝐵 ≺ ω)) |
16 | 15 | ibir 268 | . . 3 ⊢ (𝐵 ≺ ω → ({1o} × 𝐵) ≺ ω) |
17 | unfi2 9265 | . . 3 ⊢ ((({∅} × 𝐴) ≺ ω ∧ ({1o} × 𝐵) ≺ ω) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ≺ ω) | |
18 | 9, 16, 17 | syl2an 597 | . 2 ⊢ ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ≺ ω) |
19 | 1, 18 | eqbrtrid 5144 | 1 ⊢ ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴 ⊔ 𝐵) ≺ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 Vcvv 3447 ∪ cun 3912 ∅c0 4286 {csn 4590 class class class wbr 5109 × cxp 5635 Oncon0 6321 ωcom 7806 1oc1o 8409 ≈ cen 8886 ≺ csdm 8888 ⊔ 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-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
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-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-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-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-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-dju 9845 |
This theorem is referenced by: canthp1lem2 10597 |
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