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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 9996 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 9890), 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 10169 for a shorter proof that doesn't use pm54.43 9996. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
dju1p1e2 | ⊢ (1o ⊔ 1o) ≈ 2o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dju 9896 | . 2 ⊢ (1o ⊔ 1o) = (({∅} × 1o) ∪ ({1o} × 1o)) | |
2 | xp01disjl 8492 | . . 3 ⊢ (({∅} × 1o) ∩ ({1o} × 1o)) = ∅ | |
3 | 0ex 5308 | . . . . 5 ⊢ ∅ ∈ V | |
4 | 1on 8478 | . . . . 5 ⊢ 1o ∈ On | |
5 | xpsnen2g 9065 | . . . . 5 ⊢ ((∅ ∈ V ∧ 1o ∈ On) → ({∅} × 1o) ≈ 1o) | |
6 | 3, 4, 5 | mp2an 691 | . . . 4 ⊢ ({∅} × 1o) ≈ 1o |
7 | xpsnen2g 9065 | . . . . 5 ⊢ ((1o ∈ On ∧ 1o ∈ On) → ({1o} × 1o) ≈ 1o) | |
8 | 4, 4, 7 | mp2an 691 | . . . 4 ⊢ ({1o} × 1o) ≈ 1o |
9 | pm54.43 9996 | . . . 4 ⊢ ((({∅} × 1o) ≈ 1o ∧ ({1o} × 1o) ≈ 1o) → ((({∅} × 1o) ∩ ({1o} × 1o)) = ∅ ↔ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o)) | |
10 | 6, 8, 9 | mp2an 691 | . . 3 ⊢ ((({∅} × 1o) ∩ ({1o} × 1o)) = ∅ ↔ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o) |
11 | 2, 10 | mpbi 229 | . 2 ⊢ (({∅} × 1o) ∪ ({1o} × 1o)) ≈ 2o |
12 | 1, 11 | eqbrtri 5170 | 1 ⊢ (1o ⊔ 1o) ≈ 2o |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∪ cun 3947 ∩ cin 3948 ∅c0 4323 {csn 4629 class class class wbr 5149 × cxp 5675 Oncon0 6365 1oc1o 8459 2oc2o 8460 ≈ cen 8936 ⊔ cdju 9893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ord 6368 df-on 6369 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-1st 7975 df-2nd 7976 df-1o 8466 df-2o 8467 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-dju 9896 |
This theorem is referenced by: pr2dom 42278 |
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