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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 9330 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | 0elon 6244 | . . . . . 6 ⊢ ∅ ∈ On | |
3 | relsdom 8516 | . . . . . . 7 ⊢ Rel ≺ | |
4 | 3 | brrelex1i 5608 | . . . . . 6 ⊢ (𝐴 ≺ ω → 𝐴 ∈ V) |
5 | xpsnen2g 8610 | . . . . . 6 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ V) → ({∅} × 𝐴) ≈ 𝐴) | |
6 | 2, 4, 5 | sylancr 589 | . . . . 5 ⊢ (𝐴 ≺ ω → ({∅} × 𝐴) ≈ 𝐴) |
7 | sdomen1 8661 | . . . . 5 ⊢ (({∅} × 𝐴) ≈ 𝐴 → (({∅} × 𝐴) ≺ ω ↔ 𝐴 ≺ ω)) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝐴 ≺ ω → (({∅} × 𝐴) ≺ ω ↔ 𝐴 ≺ ω)) |
9 | 8 | ibir 270 | . . 3 ⊢ (𝐴 ≺ ω → ({∅} × 𝐴) ≺ ω) |
10 | 1on 8109 | . . . . . 6 ⊢ 1o ∈ On | |
11 | 3 | brrelex1i 5608 | . . . . . 6 ⊢ (𝐵 ≺ ω → 𝐵 ∈ V) |
12 | xpsnen2g 8610 | . . . . . 6 ⊢ ((1o ∈ On ∧ 𝐵 ∈ V) → ({1o} × 𝐵) ≈ 𝐵) | |
13 | 10, 11, 12 | sylancr 589 | . . . . 5 ⊢ (𝐵 ≺ ω → ({1o} × 𝐵) ≈ 𝐵) |
14 | sdomen1 8661 | . . . . 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 270 | . . 3 ⊢ (𝐵 ≺ ω → ({1o} × 𝐵) ≺ ω) |
17 | unfi2 8787 | . . 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 5101 | 1 ⊢ ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴 ⊔ 𝐵) ≺ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 Vcvv 3494 ∪ cun 3934 ∅c0 4291 {csn 4567 class class class wbr 5066 × cxp 5553 Oncon0 6191 ωcom 7580 1oc1o 8095 ≈ cen 8506 ≺ csdm 8508 ⊔ cdju 9327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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-rab 3147 df-v 3496 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 |
This theorem is referenced by: canthp1lem2 10075 |
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