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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 5298 | . . 3 ⊢ ∅ ∈ V | |
2 | in0 4384 | . . 3 ⊢ (𝐴 ∩ ∅) = ∅ | |
3 | endjudisj 10160 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V ∧ (𝐴 ∩ ∅) = ∅) → (𝐴 ⊔ ∅) ≈ (𝐴 ∪ ∅)) | |
4 | 1, 2, 3 | mp3an23 1449 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊔ ∅) ≈ (𝐴 ∪ ∅)) |
5 | un0 4383 | . 2 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
6 | 4, 5 | breqtrdi 5180 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊔ ∅) ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ∪ cun 3939 ∩ cin 3940 ∅c0 4315 class class class wbr 5139 ≈ cen 8933 ⊔ cdju 9890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-ord 6358 df-on 6359 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-1st 7969 df-2nd 7970 df-1o 8462 df-er 8700 df-en 8937 df-dju 9893 |
This theorem is referenced by: djulepw 10184 nnadju 10189 |
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