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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 8103 | . . . 4 ⊢ 1o ∈ V | |
2 | xpsnen2g 8603 | . . . 4 ⊢ ((1o ∈ V ∧ 𝐴 ∈ 𝑉) → ({1o} × 𝐴) ≈ 𝐴) | |
3 | 1, 2 | mpan 688 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({1o} × 𝐴) ≈ 𝐴) |
4 | 0ex 5204 | . . . 4 ⊢ ∅ ∈ V | |
5 | xpsnen2g 8603 | . . . 4 ⊢ ((∅ ∈ V ∧ 𝐵 ∈ 𝑊) → ({∅} × 𝐵) ≈ 𝐵) | |
6 | 4, 5 | mpan 688 | . . 3 ⊢ (𝐵 ∈ 𝑊 → ({∅} × 𝐵) ≈ 𝐵) |
7 | ensym 8551 | . . . 4 ⊢ (({1o} × 𝐴) ≈ 𝐴 → 𝐴 ≈ ({1o} × 𝐴)) | |
8 | ensym 8551 | . . . 4 ⊢ (({∅} × 𝐵) ≈ 𝐵 → 𝐵 ≈ ({∅} × 𝐵)) | |
9 | incom 4171 | . . . . . 6 ⊢ (({1o} × 𝐴) ∩ ({∅} × 𝐵)) = (({∅} × 𝐵) ∩ ({1o} × 𝐴)) | |
10 | xp01disjl 8114 | . . . . . 6 ⊢ (({∅} × 𝐵) ∩ ({1o} × 𝐴)) = ∅ | |
11 | 9, 10 | eqtri 2843 | . . . . 5 ⊢ (({1o} × 𝐴) ∩ ({∅} × 𝐵)) = ∅ |
12 | djuenun 9589 | . . . . 5 ⊢ ((𝐴 ≈ ({1o} × 𝐴) ∧ 𝐵 ≈ ({∅} × 𝐵) ∧ (({1o} × 𝐴) ∩ ({∅} × 𝐵)) = ∅) → (𝐴 ⊔ 𝐵) ≈ (({1o} × 𝐴) ∪ ({∅} × 𝐵))) | |
13 | 11, 12 | mp3an3 1445 | . . . 4 ⊢ ((𝐴 ≈ ({1o} × 𝐴) ∧ 𝐵 ≈ ({∅} × 𝐵)) → (𝐴 ⊔ 𝐵) ≈ (({1o} × 𝐴) ∪ ({∅} × 𝐵))) |
14 | 7, 8, 13 | syl2an 597 | . . 3 ⊢ ((({1o} × 𝐴) ≈ 𝐴 ∧ ({∅} × 𝐵) ≈ 𝐵) → (𝐴 ⊔ 𝐵) ≈ (({1o} × 𝐴) ∪ ({∅} × 𝐵))) |
15 | 3, 6, 14 | syl2an 597 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ≈ (({1o} × 𝐴) ∪ ({∅} × 𝐵))) |
16 | df-dju 9323 | . . 3 ⊢ (𝐵 ⊔ 𝐴) = (({∅} × 𝐵) ∪ ({1o} × 𝐴)) | |
17 | 16 | equncomi 4124 | . 2 ⊢ (𝐵 ⊔ 𝐴) = (({1o} × 𝐴) ∪ ({∅} × 𝐵)) |
18 | 15, 17 | breqtrrdi 5101 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ≈ (𝐵 ⊔ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3491 ∪ cun 3927 ∩ cin 3928 ∅c0 4284 {csn 4560 class class class wbr 5059 × cxp 5546 1oc1o 8088 ≈ cen 8499 ⊔ cdju 9320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-1st 7682 df-2nd 7683 df-1o 8095 df-er 8282 df-en 8503 df-dju 9323 |
This theorem is referenced by: djudom2 9602 djulepw 9611 infdju 9623 alephadd 9992 gchdomtri 10044 pwxpndom 10081 gchpwdom 10085 gchhar 10094 |
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