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Mirrors > Home > MPE Home > Th. List > cdacomen | 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 |
---|---|
cdacomen | ⊢ (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1on 7454 | . . . . 5 ⊢ 1𝑜 ∈ On | |
2 | xpsneng 7930 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 1𝑜 ∈ On) → (𝐴 × {1𝑜}) ≈ 𝐴) | |
3 | 1, 2 | mpan2 703 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 × {1𝑜}) ≈ 𝐴) |
4 | 0ex 4718 | . . . . 5 ⊢ ∅ ∈ V | |
5 | xpsneng 7930 | . . . . 5 ⊢ ((𝐵 ∈ V ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵) | |
6 | 4, 5 | mpan2 703 | . . . 4 ⊢ (𝐵 ∈ V → (𝐵 × {∅}) ≈ 𝐵) |
7 | ensym 7891 | . . . . 5 ⊢ ((𝐴 × {1𝑜}) ≈ 𝐴 → 𝐴 ≈ (𝐴 × {1𝑜})) | |
8 | ensym 7891 | . . . . 5 ⊢ ((𝐵 × {∅}) ≈ 𝐵 → 𝐵 ≈ (𝐵 × {∅})) | |
9 | incom 3767 | . . . . . . 7 ⊢ ((𝐴 × {1𝑜}) ∩ (𝐵 × {∅})) = ((𝐵 × {∅}) ∩ (𝐴 × {1𝑜})) | |
10 | xp01disj 7463 | . . . . . . 7 ⊢ ((𝐵 × {∅}) ∩ (𝐴 × {1𝑜})) = ∅ | |
11 | 9, 10 | eqtri 2632 | . . . . . 6 ⊢ ((𝐴 × {1𝑜}) ∩ (𝐵 × {∅})) = ∅ |
12 | cdaenun 8879 | . . . . . 6 ⊢ ((𝐴 ≈ (𝐴 × {1𝑜}) ∧ 𝐵 ≈ (𝐵 × {∅}) ∧ ((𝐴 × {1𝑜}) ∩ (𝐵 × {∅})) = ∅) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅}))) | |
13 | 11, 12 | mp3an3 1405 | . . . . 5 ⊢ ((𝐴 ≈ (𝐴 × {1𝑜}) ∧ 𝐵 ≈ (𝐵 × {∅})) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅}))) |
14 | 7, 8, 13 | syl2an 493 | . . . 4 ⊢ (((𝐴 × {1𝑜}) ≈ 𝐴 ∧ (𝐵 × {∅}) ≈ 𝐵) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅}))) |
15 | 3, 6, 14 | syl2an 493 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅}))) |
16 | cdaval 8875 | . . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐵 × {∅}) ∪ (𝐴 × {1𝑜}))) | |
17 | 16 | ancoms 468 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐵 × {∅}) ∪ (𝐴 × {1𝑜}))) |
18 | uncom 3719 | . . . 4 ⊢ ((𝐵 × {∅}) ∪ (𝐴 × {1𝑜})) = ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅})) | |
19 | 17, 18 | syl6eq 2660 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ((𝐴 × {1𝑜}) ∪ (𝐵 × {∅}))) |
20 | 15, 19 | breqtrrd 4611 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴)) |
21 | 4 | enref 7874 | . . . 4 ⊢ ∅ ≈ ∅ |
22 | 21 | a1i 11 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∅ ≈ ∅) |
23 | cdafn 8874 | . . . . 5 ⊢ +𝑐 Fn (V × V) | |
24 | fndm 5904 | . . . . 5 ⊢ ( +𝑐 Fn (V × V) → dom +𝑐 = (V × V)) | |
25 | 23, 24 | ax-mp 5 | . . . 4 ⊢ dom +𝑐 = (V × V) |
26 | 25 | ndmov 6716 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) = ∅) |
27 | ancom 465 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V)) | |
28 | 25 | ndmov 6716 | . . . 4 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 +𝑐 𝐴) = ∅) |
29 | 27, 28 | sylnbi 319 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 +𝑐 𝐴) = ∅) |
30 | 22, 26, 29 | 3brtr4d 4615 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴)) |
31 | 20, 30 | pm2.61i 175 | 1 ⊢ (𝐴 +𝑐 𝐵) ≈ (𝐵 +𝑐 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 {csn 4125 class class class wbr 4583 × cxp 5036 dom cdm 5038 Oncon0 5640 Fn wfn 5799 (class class class)co 6549 1𝑜c1o 7440 ≈ cen 7838 +𝑐 ccda 8872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-1o 7447 df-er 7629 df-en 7842 df-cda 8873 |
This theorem is referenced by: cdadom2 8892 cdalepw 8901 infcda 8913 alephadd 9278 gchdomtri 9330 pwxpndom 9367 gchpwdom 9371 gchhar 9380 |
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