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Mirrors > Home > MPE Home > Th. List > cdaenun | Structured version Visualization version GIF version |
Description: Cardinal addition is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
cdaenun | ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 +𝑐 𝐶) ≈ (𝐵 ∪ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdaen 8878 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 +𝑐 𝐶) ≈ (𝐵 +𝑐 𝐷)) | |
2 | 1 | 3adant3 1074 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 +𝑐 𝐶) ≈ (𝐵 +𝑐 𝐷)) |
3 | relen 7846 | . . . 4 ⊢ Rel ≈ | |
4 | 3 | brrelex2i 5083 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V) |
5 | 3 | brrelex2i 5083 | . . 3 ⊢ (𝐶 ≈ 𝐷 → 𝐷 ∈ V) |
6 | id 22 | . . 3 ⊢ ((𝐵 ∩ 𝐷) = ∅ → (𝐵 ∩ 𝐷) = ∅) | |
7 | cdaun 8877 | . . 3 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐵 +𝑐 𝐷) ≈ (𝐵 ∪ 𝐷)) | |
8 | 4, 5, 6, 7 | syl3an 1360 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐵 +𝑐 𝐷) ≈ (𝐵 ∪ 𝐷)) |
9 | entr 7894 | . 2 ⊢ (((𝐴 +𝑐 𝐶) ≈ (𝐵 +𝑐 𝐷) ∧ (𝐵 +𝑐 𝐷) ≈ (𝐵 ∪ 𝐷)) → (𝐴 +𝑐 𝐶) ≈ (𝐵 ∪ 𝐷)) | |
10 | 2, 8, 9 | syl2anc 691 | 1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 +𝑐 𝐶) ≈ (𝐵 ∪ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 class class class wbr 4583 (class class class)co 6549 ≈ 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-res 5050 df-ima 5051 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-er 7629 df-en 7842 df-cda 8873 |
This theorem is referenced by: cda1en 8880 cdacomen 8886 cdaassen 8887 xpcdaen 8888 onacda 8902 pwxpndom2 9366 |
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