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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 5307 | . . 3 ⊢ ∅ ∈ V | |
2 | in0 4391 | . . 3 ⊢ (𝐴 ∩ ∅) = ∅ | |
3 | endjudisj 10162 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V ∧ (𝐴 ∩ ∅) = ∅) → (𝐴 ⊔ ∅) ≈ (𝐴 ∪ ∅)) | |
4 | 1, 2, 3 | mp3an23 1453 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊔ ∅) ≈ (𝐴 ∪ ∅)) |
5 | un0 4390 | . 2 ⊢ (𝐴 ∪ ∅) = 𝐴 | |
6 | 4, 5 | breqtrdi 5189 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊔ ∅) ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∪ cun 3946 ∩ cin 3947 ∅c0 4322 class class class wbr 5148 ≈ cen 8935 ⊔ cdju 9892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-1st 7974 df-2nd 7975 df-1o 8465 df-er 8702 df-en 8939 df-dju 9895 |
This theorem is referenced by: djulepw 10186 nnadju 10191 |
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