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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 9323 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | 0elon 6237 | . . . . . 6 ⊢ ∅ ∈ On | |
3 | relsdom 8509 | . . . . . . 7 ⊢ Rel ≺ | |
4 | 3 | brrelex1i 5601 | . . . . . 6 ⊢ (𝐴 ≺ ω → 𝐴 ∈ V) |
5 | xpsnen2g 8603 | . . . . . 6 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ V) → ({∅} × 𝐴) ≈ 𝐴) | |
6 | 2, 4, 5 | sylancr 589 | . . . . 5 ⊢ (𝐴 ≺ ω → ({∅} × 𝐴) ≈ 𝐴) |
7 | sdomen1 8654 | . . . . 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 8102 | . . . . . 6 ⊢ 1o ∈ On | |
11 | 3 | brrelex1i 5601 | . . . . . 6 ⊢ (𝐵 ≺ ω → 𝐵 ∈ V) |
12 | xpsnen2g 8603 | . . . . . 6 ⊢ ((1o ∈ On ∧ 𝐵 ∈ V) → ({1o} × 𝐵) ≈ 𝐵) | |
13 | 10, 11, 12 | sylancr 589 | . . . . 5 ⊢ (𝐵 ≺ ω → ({1o} × 𝐵) ≈ 𝐵) |
14 | sdomen1 8654 | . . . . 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 8780 | . . 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 5094 | 1 ⊢ ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴 ⊔ 𝐵) ≺ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2113 Vcvv 3491 ∪ cun 3927 ∅c0 4284 {csn 4560 class class class wbr 5059 × cxp 5546 Oncon0 6184 ωcom 7573 1oc1o 8088 ≈ cen 8499 ≺ csdm 8501 ⊔ cdju 9320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 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-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-dju 9323 |
This theorem is referenced by: canthp1lem2 10068 |
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