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Mirrors > Home > MPE Home > Th. List > pm110.643 | Structured version Visualization version GIF version |
Description: 1+1=2 for cardinal number addition, derived from pm54.43 8709 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 8641), but after applying definitions, our theorem is equivalent. The comment for cdaval 8875 explains why we use ≈ instead of =. See pm110.643ALT 8883 for a shorter proof that doesn't use pm54.43 8709. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
pm110.643 | ⊢ (1𝑜 +𝑐 1𝑜) ≈ 2𝑜 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1on 7454 | . . 3 ⊢ 1𝑜 ∈ On | |
2 | cdaval 8875 | . . 3 ⊢ ((1𝑜 ∈ On ∧ 1𝑜 ∈ On) → (1𝑜 +𝑐 1𝑜) = ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜}))) | |
3 | 1, 1, 2 | mp2an 704 | . 2 ⊢ (1𝑜 +𝑐 1𝑜) = ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) |
4 | xp01disj 7463 | . . 3 ⊢ ((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ | |
5 | 1 | elexi 3186 | . . . . 5 ⊢ 1𝑜 ∈ V |
6 | 0ex 4718 | . . . . 5 ⊢ ∅ ∈ V | |
7 | 5, 6 | xpsnen 7929 | . . . 4 ⊢ (1𝑜 × {∅}) ≈ 1𝑜 |
8 | 5, 5 | xpsnen 7929 | . . . 4 ⊢ (1𝑜 × {1𝑜}) ≈ 1𝑜 |
9 | pm54.43 8709 | . . . 4 ⊢ (((1𝑜 × {∅}) ≈ 1𝑜 ∧ (1𝑜 × {1𝑜}) ≈ 1𝑜) → (((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜)) | |
10 | 7, 8, 9 | mp2an 704 | . . 3 ⊢ (((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜) |
11 | 4, 10 | mpbi 219 | . 2 ⊢ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜 |
12 | 3, 11 | eqbrtri 4604 | 1 ⊢ (1𝑜 +𝑐 1𝑜) ≈ 2𝑜 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 {csn 4125 class class class wbr 4583 × cxp 5036 Oncon0 5640 (class class class)co 6549 1𝑜c1o 7440 2𝑜c2o 7441 ≈ cen 7838 +𝑐 ccda 8872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1o 7447 df-2o 7448 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-cda 8873 |
This theorem is referenced by: (None) |
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