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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 10039 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 9933), 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 10213 for a shorter proof that doesn't use pm54.43 10039. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
dju1p1e2 | ⊢ (1o ⊔ 1o) ≈ 2o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dju 9939 | . 2 ⊢ (1o ⊔ 1o) = (({∅} × 1o) ∪ ({1o} × 1o)) | |
2 | xp01disjl 8529 | . . 3 ⊢ (({∅} × 1o) ∩ ({1o} × 1o)) = ∅ | |
3 | 0ex 5313 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 1on 8517 | . . . . 5 ⊢ 1o ∈ On | |
5 | xpsnen2g 9104 | . . . . 5 ⊢ ((∅ ∈ V ∧ 1o ∈ On) → ({∅} × 1o) ≈ 1o) | |
6 | 3, 4, 5 | mp2an 692 | . . . 4 ⊢ ({∅} × 1o) ≈ 1o |
7 | xpsnen2g 9104 | . . . . 5 ⊢ ((1o ∈ On ∧ 1o ∈ On) → ({1o} × 1o) ≈ 1o) | |
8 | 4, 4, 7 | mp2an 692 | . . . 4 ⊢ ({1o} × 1o) ≈ 1o |
9 | pm54.43 10039 | . . . 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 5169 | 1 ⊢ (1o ⊔ 1o) ≈ 2o |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 ∩ cin 3962 ∅c0 4339 {csn 4631 class class class wbr 5148 × cxp 5687 Oncon0 6386 1oc1o 8498 2oc2o 8499 ≈ cen 8981 ⊔ cdju 9936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-1st 8013 df-2nd 8014 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-dju 9939 |
This theorem is referenced by: pr2dom 43517 |
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