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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 8407 | . . . 4 ⊢ 1o ∈ V | |
| 2 | xpsnen2g 8998 | . . . 4 ⊢ ((1o ∈ V ∧ 𝐴 ∈ 𝑉) → ({1o} × 𝐴) ≈ 𝐴) | |
| 3 | 1, 2 | mpan 690 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({1o} × 𝐴) ≈ 𝐴) |
| 4 | 0ex 5252 | . . . 4 ⊢ ∅ ∈ V | |
| 5 | xpsnen2g 8998 | . . . 4 ⊢ ((∅ ∈ V ∧ 𝐵 ∈ 𝑊) → ({∅} × 𝐵) ≈ 𝐵) | |
| 6 | 4, 5 | mpan 690 | . . 3 ⊢ (𝐵 ∈ 𝑊 → ({∅} × 𝐵) ≈ 𝐵) |
| 7 | ensym 8940 | . . . 4 ⊢ (({1o} × 𝐴) ≈ 𝐴 → 𝐴 ≈ ({1o} × 𝐴)) | |
| 8 | ensym 8940 | . . . 4 ⊢ (({∅} × 𝐵) ≈ 𝐵 → 𝐵 ≈ ({∅} × 𝐵)) | |
| 9 | incom 4161 | . . . . . 6 ⊢ (({1o} × 𝐴) ∩ ({∅} × 𝐵)) = (({∅} × 𝐵) ∩ ({1o} × 𝐴)) | |
| 10 | xp01disjl 8419 | . . . . . 6 ⊢ (({∅} × 𝐵) ∩ ({1o} × 𝐴)) = ∅ | |
| 11 | 9, 10 | eqtri 2759 | . . . . 5 ⊢ (({1o} × 𝐴) ∩ ({∅} × 𝐵)) = ∅ |
| 12 | djuenun 10081 | . . . . 5 ⊢ ((𝐴 ≈ ({1o} × 𝐴) ∧ 𝐵 ≈ ({∅} × 𝐵) ∧ (({1o} × 𝐴) ∩ ({∅} × 𝐵)) = ∅) → (𝐴 ⊔ 𝐵) ≈ (({1o} × 𝐴) ∪ ({∅} × 𝐵))) | |
| 13 | 11, 12 | mp3an3 1452 | . . . 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 9813 | . . 3 ⊢ (𝐵 ⊔ 𝐴) = (({∅} × 𝐵) ∪ ({1o} × 𝐴)) | |
| 17 | 16 | equncomi 4112 | . 2 ⊢ (𝐵 ⊔ 𝐴) = (({1o} × 𝐴) ∪ ({∅} × 𝐵)) |
| 18 | 15, 17 | breqtrrdi 5140 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ≈ (𝐵 ⊔ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 Vcvv 3440 ∪ cun 3899 ∩ cin 3900 ∅c0 4285 {csn 4580 class class class wbr 5098 × cxp 5622 1oc1o 8390 ≈ cen 8880 ⊔ cdju 9810 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-ord 6320 df-on 6321 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-1st 7933 df-2nd 7934 df-1o 8397 df-er 8635 df-en 8884 df-dju 9813 |
| This theorem is referenced by: djudom2 10094 djulepw 10103 infdju 10117 alephadd 10488 gchdomtri 10540 pwxpndom 10577 gchpwdom 10581 gchhar 10590 |
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