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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 8496 | . . . 4 ⊢ 1o ∈ V | |
2 | xpsnen2g 9089 | . . . 4 ⊢ ((1o ∈ V ∧ 𝐴 ∈ 𝑉) → ({1o} × 𝐴) ≈ 𝐴) | |
3 | 1, 2 | mpan 689 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({1o} × 𝐴) ≈ 𝐴) |
4 | 0ex 5307 | . . . 4 ⊢ ∅ ∈ V | |
5 | xpsnen2g 9089 | . . . 4 ⊢ ((∅ ∈ V ∧ 𝐵 ∈ 𝑊) → ({∅} × 𝐵) ≈ 𝐵) | |
6 | 4, 5 | mpan 689 | . . 3 ⊢ (𝐵 ∈ 𝑊 → ({∅} × 𝐵) ≈ 𝐵) |
7 | ensym 9023 | . . . 4 ⊢ (({1o} × 𝐴) ≈ 𝐴 → 𝐴 ≈ ({1o} × 𝐴)) | |
8 | ensym 9023 | . . . 4 ⊢ (({∅} × 𝐵) ≈ 𝐵 → 𝐵 ≈ ({∅} × 𝐵)) | |
9 | incom 4201 | . . . . . 6 ⊢ (({1o} × 𝐴) ∩ ({∅} × 𝐵)) = (({∅} × 𝐵) ∩ ({1o} × 𝐴)) | |
10 | xp01disjl 8512 | . . . . . 6 ⊢ (({∅} × 𝐵) ∩ ({1o} × 𝐴)) = ∅ | |
11 | 9, 10 | eqtri 2756 | . . . . 5 ⊢ (({1o} × 𝐴) ∩ ({∅} × 𝐵)) = ∅ |
12 | djuenun 10193 | . . . . 5 ⊢ ((𝐴 ≈ ({1o} × 𝐴) ∧ 𝐵 ≈ ({∅} × 𝐵) ∧ (({1o} × 𝐴) ∩ ({∅} × 𝐵)) = ∅) → (𝐴 ⊔ 𝐵) ≈ (({1o} × 𝐴) ∪ ({∅} × 𝐵))) | |
13 | 11, 12 | mp3an3 1447 | . . . 4 ⊢ ((𝐴 ≈ ({1o} × 𝐴) ∧ 𝐵 ≈ ({∅} × 𝐵)) → (𝐴 ⊔ 𝐵) ≈ (({1o} × 𝐴) ∪ ({∅} × 𝐵))) |
14 | 7, 8, 13 | syl2an 595 | . . 3 ⊢ ((({1o} × 𝐴) ≈ 𝐴 ∧ ({∅} × 𝐵) ≈ 𝐵) → (𝐴 ⊔ 𝐵) ≈ (({1o} × 𝐴) ∪ ({∅} × 𝐵))) |
15 | 3, 6, 14 | syl2an 595 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ≈ (({1o} × 𝐴) ∪ ({∅} × 𝐵))) |
16 | df-dju 9924 | . . 3 ⊢ (𝐵 ⊔ 𝐴) = (({∅} × 𝐵) ∪ ({1o} × 𝐴)) | |
17 | 16 | equncomi 4154 | . 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 1534 ∈ wcel 2099 Vcvv 3471 ∪ cun 3945 ∩ cin 3946 ∅c0 4323 {csn 4629 class class class wbr 5148 × cxp 5676 1oc1o 8479 ≈ cen 8960 ⊔ cdju 9921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-1st 7993 df-2nd 7994 df-1o 8486 df-er 8724 df-en 8964 df-dju 9924 |
This theorem is referenced by: djudom2 10206 djulepw 10215 infdju 10231 alephadd 10600 gchdomtri 10652 pwxpndom 10689 gchpwdom 10693 gchhar 10702 |
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