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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 8995 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 +𝑐 𝐶) ≈ (𝐵 +𝑐 𝐷)) | |
2 | 1 | 3adant3 1081 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 +𝑐 𝐶) ≈ (𝐵 +𝑐 𝐷)) |
3 | relen 7960 | . . . 4 ⊢ Rel ≈ | |
4 | 3 | brrelex2i 5159 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V) |
5 | 3 | brrelex2i 5159 | . . 3 ⊢ (𝐶 ≈ 𝐷 → 𝐷 ∈ V) |
6 | id 22 | . . 3 ⊢ ((𝐵 ∩ 𝐷) = ∅ → (𝐵 ∩ 𝐷) = ∅) | |
7 | cdaun 8994 | . . 3 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐵 +𝑐 𝐷) ≈ (𝐵 ∪ 𝐷)) | |
8 | 4, 5, 6, 7 | syl3an 1368 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐵 +𝑐 𝐷) ≈ (𝐵 ∪ 𝐷)) |
9 | entr 8008 | . 2 ⊢ (((𝐴 +𝑐 𝐶) ≈ (𝐵 +𝑐 𝐷) ∧ (𝐵 +𝑐 𝐷) ≈ (𝐵 ∪ 𝐷)) → (𝐴 +𝑐 𝐶) ≈ (𝐵 ∪ 𝐷)) | |
10 | 2, 8, 9 | syl2anc 693 | 1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 +𝑐 𝐶) ≈ (𝐵 ∪ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∪ cun 3572 ∩ cin 3573 ∅c0 3915 class class class wbr 4653 (class class class)co 6650 ≈ cen 7952 +𝑐 ccda 8989 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1o 7560 df-er 7742 df-en 7956 df-cda 8990 |
This theorem is referenced by: cda1en 8997 cdacomen 9003 cdaassen 9004 xpcdaen 9005 onacda 9019 pwxpndom2 9487 |
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