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Mirrors > Home > MPE Home > Th. List > cdaun | Structured version Visualization version GIF version |
Description: Cardinal addition is equinumerous to union for disjoint sets. (Contributed by NM, 5-Apr-2007.) |
Ref | Expression |
---|---|
cdaun | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 +𝑐 𝐵) ≈ (𝐴 ∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdaval 8875 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜}))) | |
2 | 1 | 3adant3 1074 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 +𝑐 𝐵) = ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜}))) |
3 | 0ex 4718 | . . . . . 6 ⊢ ∅ ∈ V | |
4 | xpsneng 7930 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → (𝐴 × {∅}) ≈ 𝐴) | |
5 | 3, 4 | mpan2 703 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {∅}) ≈ 𝐴) |
6 | 1on 7454 | . . . . . 6 ⊢ 1𝑜 ∈ On | |
7 | xpsneng 7930 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ 1𝑜 ∈ On) → (𝐵 × {1𝑜}) ≈ 𝐵) | |
8 | 6, 7 | mpan2 703 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → (𝐵 × {1𝑜}) ≈ 𝐵) |
9 | 5, 8 | anim12i 588 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵)) |
10 | xp01disj 7463 | . . . . 5 ⊢ ((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅ | |
11 | 10 | jctl 562 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅ ∧ (𝐴 ∩ 𝐵) = ∅)) |
12 | unen 7925 | . . . 4 ⊢ ((((𝐴 × {∅}) ≈ 𝐴 ∧ (𝐵 × {1𝑜}) ≈ 𝐵) ∧ (((𝐴 × {∅}) ∩ (𝐵 × {1𝑜})) = ∅ ∧ (𝐴 ∩ 𝐵) = ∅)) → ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ≈ (𝐴 ∪ 𝐵)) | |
13 | 9, 11, 12 | syl2an 493 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ≈ (𝐴 ∪ 𝐵)) |
14 | 13 | 3impa 1251 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 × {∅}) ∪ (𝐵 × {1𝑜})) ≈ (𝐴 ∪ 𝐵)) |
15 | 2, 14 | eqbrtrd 4605 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 +𝑐 𝐵) ≈ (𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 {csn 4125 class class class wbr 4583 × cxp 5036 Oncon0 5640 (class class class)co 6549 1𝑜c1o 7440 ≈ cen 7838 +𝑐 ccda 8872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1o 7447 df-en 7842 df-cda 8873 |
This theorem is referenced by: cdaenun 8879 cda0en 8884 ficardun 8907 ackbij1lem9 8933 canthp1lem1 9353 |
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