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Mirrors > Home > MPE Home > Th. List > djucomen | Structured version Visualization version GIF version |
Description: Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
djucomen | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ≈ (𝐵 ⊔ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1oex 8515 | . . . 4 ⊢ 1o ∈ V | |
2 | xpsnen2g 9104 | . . . 4 ⊢ ((1o ∈ V ∧ 𝐴 ∈ 𝑉) → ({1o} × 𝐴) ≈ 𝐴) | |
3 | 1, 2 | mpan 690 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({1o} × 𝐴) ≈ 𝐴) |
4 | 0ex 5313 | . . . 4 ⊢ ∅ ∈ V | |
5 | xpsnen2g 9104 | . . . 4 ⊢ ((∅ ∈ V ∧ 𝐵 ∈ 𝑊) → ({∅} × 𝐵) ≈ 𝐵) | |
6 | 4, 5 | mpan 690 | . . 3 ⊢ (𝐵 ∈ 𝑊 → ({∅} × 𝐵) ≈ 𝐵) |
7 | ensym 9042 | . . . 4 ⊢ (({1o} × 𝐴) ≈ 𝐴 → 𝐴 ≈ ({1o} × 𝐴)) | |
8 | ensym 9042 | . . . 4 ⊢ (({∅} × 𝐵) ≈ 𝐵 → 𝐵 ≈ ({∅} × 𝐵)) | |
9 | incom 4217 | . . . . . 6 ⊢ (({1o} × 𝐴) ∩ ({∅} × 𝐵)) = (({∅} × 𝐵) ∩ ({1o} × 𝐴)) | |
10 | xp01disjl 8529 | . . . . . 6 ⊢ (({∅} × 𝐵) ∩ ({1o} × 𝐴)) = ∅ | |
11 | 9, 10 | eqtri 2763 | . . . . 5 ⊢ (({1o} × 𝐴) ∩ ({∅} × 𝐵)) = ∅ |
12 | djuenun 10209 | . . . . 5 ⊢ ((𝐴 ≈ ({1o} × 𝐴) ∧ 𝐵 ≈ ({∅} × 𝐵) ∧ (({1o} × 𝐴) ∩ ({∅} × 𝐵)) = ∅) → (𝐴 ⊔ 𝐵) ≈ (({1o} × 𝐴) ∪ ({∅} × 𝐵))) | |
13 | 11, 12 | mp3an3 1449 | . . . 4 ⊢ ((𝐴 ≈ ({1o} × 𝐴) ∧ 𝐵 ≈ ({∅} × 𝐵)) → (𝐴 ⊔ 𝐵) ≈ (({1o} × 𝐴) ∪ ({∅} × 𝐵))) |
14 | 7, 8, 13 | syl2an 596 | . . 3 ⊢ ((({1o} × 𝐴) ≈ 𝐴 ∧ ({∅} × 𝐵) ≈ 𝐵) → (𝐴 ⊔ 𝐵) ≈ (({1o} × 𝐴) ∪ ({∅} × 𝐵))) |
15 | 3, 6, 14 | syl2an 596 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ≈ (({1o} × 𝐴) ∪ ({∅} × 𝐵))) |
16 | df-dju 9939 | . . 3 ⊢ (𝐵 ⊔ 𝐴) = (({∅} × 𝐵) ∪ ({1o} × 𝐴)) | |
17 | 16 | equncomi 4170 | . 2 ⊢ (𝐵 ⊔ 𝐴) = (({1o} × 𝐴) ∪ ({∅} × 𝐵)) |
18 | 15, 17 | breqtrrdi 5190 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ≈ (𝐵 ⊔ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 ∩ cin 3962 ∅c0 4339 {csn 4631 class class class wbr 5148 × cxp 5687 1oc1o 8498 ≈ cen 8981 ⊔ cdju 9936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-1st 8013 df-2nd 8014 df-1o 8505 df-er 8744 df-en 8985 df-dju 9939 |
This theorem is referenced by: djudom2 10222 djulepw 10231 infdju 10245 alephadd 10615 gchdomtri 10667 pwxpndom 10704 gchpwdom 10708 gchhar 10717 |
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