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| Mirrors > Home > MPE Home > Th. List > dju0en | Structured version Visualization version GIF version | ||
| Description: Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) | 
| Ref | Expression | 
|---|---|
| dju0en | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊔ ∅) ≈ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0ex 5306 | . . 3 ⊢ ∅ ∈ V | |
| 2 | in0 4394 | . . 3 ⊢ (𝐴 ∩ ∅) = ∅ | |
| 3 | endjudisj 10210 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V ∧ (𝐴 ∩ ∅) = ∅) → (𝐴 ⊔ ∅) ≈ (𝐴 ∪ ∅)) | |
| 4 | 1, 2, 3 | mp3an23 1454 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊔ ∅) ≈ (𝐴 ∪ ∅)) | 
| 5 | un0 4393 | . 2 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
| 6 | 4, 5 | breqtrdi 5183 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊔ ∅) ≈ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∪ cun 3948 ∩ cin 3949 ∅c0 4332 class class class wbr 5142 ≈ cen 8983 ⊔ cdju 9939 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6386 df-on 6387 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-1st 8015 df-2nd 8016 df-1o 8507 df-er 8746 df-en 8987 df-dju 9942 | 
| This theorem is referenced by: djulepw 10234 nnadju 10239 | 
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