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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 9998 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 9892), 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 10171 for a shorter proof that doesn't use pm54.43 9998. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
dju1p1e2 | ⊢ (1o ⊔ 1o) ≈ 2o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dju 9898 | . 2 ⊢ (1o ⊔ 1o) = (({∅} × 1o) ∪ ({1o} × 1o)) | |
2 | xp01disjl 8494 | . . 3 ⊢ (({∅} × 1o) ∩ ({1o} × 1o)) = ∅ | |
3 | 0ex 5306 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 1on 8480 | . . . . 5 ⊢ 1o ∈ On | |
5 | xpsnen2g 9067 | . . . . 5 ⊢ ((∅ ∈ V ∧ 1o ∈ On) → ({∅} × 1o) ≈ 1o) | |
6 | 3, 4, 5 | mp2an 688 | . . . 4 ⊢ ({∅} × 1o) ≈ 1o |
7 | xpsnen2g 9067 | . . . . 5 ⊢ ((1o ∈ On ∧ 1o ∈ On) → ({1o} × 1o) ≈ 1o) | |
8 | 4, 4, 7 | mp2an 688 | . . . 4 ⊢ ({1o} × 1o) ≈ 1o |
9 | pm54.43 9998 | . . . 4 ⊢ ((({∅} × 1o) ≈ 1o ∧ ({1o} × 1o) ≈ 1o) → ((({∅} × 1o) ∩ ({1o} × 1o)) = ∅ ↔ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o)) | |
10 | 6, 8, 9 | mp2an 688 | . . 3 ⊢ ((({∅} × 1o) ∩ ({1o} × 1o)) = ∅ ↔ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o) |
11 | 2, 10 | mpbi 229 | . 2 ⊢ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o |
12 | 1, 11 | eqbrtri 5168 | 1 ⊢ (1o ⊔ 1o) ≈ 2o |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1539 ∈ wcel 2104 Vcvv 3472 ∪ cun 3945 ∩ cin 3946 ∅c0 4321 {csn 4627 class class class wbr 5147 × cxp 5673 Oncon0 6363 1oc1o 8461 2oc2o 8462 ≈ cen 8938 ⊔ cdju 9895 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6366 df-on 6367 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-1st 7977 df-2nd 7978 df-1o 8468 df-2o 8469 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-dju 9898 |
This theorem is referenced by: pr2dom 42580 |
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