Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > endjudisj | Structured version Visualization version GIF version |
Description: Equinumerosity of a disjoint union and a union of two disjoint sets. (Contributed by NM, 5-Apr-2007.) |
Ref | Expression |
---|---|
endjudisj | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dju 9323 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | 0ex 5204 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | xpsnen2g 8603 | . . . . . 6 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → ({∅} × 𝐴) ≈ 𝐴) | |
4 | 2, 3 | mpan 688 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ({∅} × 𝐴) ≈ 𝐴) |
5 | 1on 8102 | . . . . . 6 ⊢ 1o ∈ On | |
6 | xpsnen2g 8603 | . . . . . 6 ⊢ ((1o ∈ On ∧ 𝐵 ∈ 𝑊) → ({1o} × 𝐵) ≈ 𝐵) | |
7 | 5, 6 | mpan 688 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → ({1o} × 𝐵) ≈ 𝐵) |
8 | 4, 7 | anim12i 614 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (({∅} × 𝐴) ≈ 𝐴 ∧ ({1o} × 𝐵) ≈ 𝐵)) |
9 | xp01disjl 8114 | . . . . 5 ⊢ (({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅ | |
10 | 9 | jctl 526 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅ ∧ (𝐴 ∩ 𝐵) = ∅)) |
11 | unen 8589 | . . . 4 ⊢ (((({∅} × 𝐴) ≈ 𝐴 ∧ ({1o} × 𝐵) ≈ 𝐵) ∧ ((({∅} × 𝐴) ∩ ({1o} × 𝐵)) = ∅ ∧ (𝐴 ∩ 𝐵) = ∅)) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ≈ (𝐴 ∪ 𝐵)) | |
12 | 8, 10, 11 | syl2an 597 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐴 ∩ 𝐵) = ∅) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ≈ (𝐴 ∪ 𝐵)) |
13 | 12 | 3impa 1105 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ≈ (𝐴 ∪ 𝐵)) |
14 | 1, 13 | eqbrtrid 5094 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 Vcvv 3491 ∪ cun 3927 ∩ cin 3928 ∅c0 4284 {csn 4560 class class class wbr 5059 × cxp 5546 Oncon0 6184 1oc1o 8088 ≈ cen 8499 ⊔ 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-rab 3146 df-v 3493 df-sbc 3769 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-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-ord 6187 df-on 6188 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-1st 7682 df-2nd 7683 df-1o 8095 df-er 8282 df-en 8503 df-dju 9323 |
This theorem is referenced by: djuenun 9589 dju0en 9594 ficardun 9617 ackbij1lem9 9643 canthp1lem1 10067 |
Copyright terms: Public domain | W3C validator |