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| Mirrors > Home > MPE Home > Th. List > dju1p1e2 | Structured version Visualization version GIF version | ||
| Description: 1+1=2 for cardinal number addition, derived from pm54.43 10041 as promised. Theorem *110.643 of Principia Mathematica, vol. II, p. 86, which adds the remark, "The above proposition is occasionally useful." Whitehead and Russell define cardinal addition on collections of all sets equinumerous to 1 and 2 (which for us are proper classes unless we restrict them as in karden 9935), but after applying definitions, our theorem is equivalent. Because we use a disjoint union for cardinal addition (as explained in the comment at the top of this section), we use ≈ instead of =. See dju1p1e2ALT 10215 for a shorter proof that doesn't use pm54.43 10041. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| dju1p1e2 | ⊢ (1o ⊔ 1o) ≈ 2o | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-dju 9941 | . 2 ⊢ (1o ⊔ 1o) = (({∅} × 1o) ∪ ({1o} × 1o)) | |
| 2 | xp01disjl 8530 | . . 3 ⊢ (({∅} × 1o) ∩ ({1o} × 1o)) = ∅ | |
| 3 | 0ex 5307 | . . . . 5 ⊢ ∅ ∈ V | |
| 4 | 1on 8518 | . . . . 5 ⊢ 1o ∈ On | |
| 5 | xpsnen2g 9105 | . . . . 5 ⊢ ((∅ ∈ V ∧ 1o ∈ On) → ({∅} × 1o) ≈ 1o) | |
| 6 | 3, 4, 5 | mp2an 692 | . . . 4 ⊢ ({∅} × 1o) ≈ 1o | 
| 7 | xpsnen2g 9105 | . . . . 5 ⊢ ((1o ∈ On ∧ 1o ∈ On) → ({1o} × 1o) ≈ 1o) | |
| 8 | 4, 4, 7 | mp2an 692 | . . . 4 ⊢ ({1o} × 1o) ≈ 1o | 
| 9 | pm54.43 10041 | . . . 4 ⊢ ((({∅} × 1o) ≈ 1o ∧ ({1o} × 1o) ≈ 1o) → ((({∅} × 1o) ∩ ({1o} × 1o)) = ∅ ↔ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o)) | |
| 10 | 6, 8, 9 | mp2an 692 | . . 3 ⊢ ((({∅} × 1o) ∩ ({1o} × 1o)) = ∅ ↔ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o) | 
| 11 | 2, 10 | mpbi 230 | . 2 ⊢ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o | 
| 12 | 1, 11 | eqbrtri 5164 | 1 ⊢ (1o ⊔ 1o) ≈ 2o | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 {csn 4626 class class class wbr 5143 × cxp 5683 Oncon0 6384 1oc1o 8499 2oc2o 8500 ≈ cen 8982 ⊔ cdju 9938 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-1st 8014 df-2nd 8015 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-dju 9941 | 
| This theorem is referenced by: pr2dom 43540 | 
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