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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 9025 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 8921), but after applying definitions, our theorem is equivalent. The comment for cdaval 9193 explains why we use ≈ instead of =. See pm110.643ALT 9201 for a shorter proof that doesn't use pm54.43 9025. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
pm110.643 | ⊢ (1𝑜 +𝑐 1𝑜) ≈ 2𝑜 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1on 7719 | . . 3 ⊢ 1𝑜 ∈ On | |
2 | cdaval 9193 | . . 3 ⊢ ((1𝑜 ∈ On ∧ 1𝑜 ∈ On) → (1𝑜 +𝑐 1𝑜) = ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜}))) | |
3 | 1, 1, 2 | mp2an 664 | . 2 ⊢ (1𝑜 +𝑐 1𝑜) = ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) |
4 | xp01disj 7729 | . . 3 ⊢ ((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ | |
5 | 1 | elexi 3364 | . . . . 5 ⊢ 1𝑜 ∈ V |
6 | 0ex 4924 | . . . . 5 ⊢ ∅ ∈ V | |
7 | 5, 6 | xpsnen 8199 | . . . 4 ⊢ (1𝑜 × {∅}) ≈ 1𝑜 |
8 | 5, 5 | xpsnen 8199 | . . . 4 ⊢ (1𝑜 × {1𝑜}) ≈ 1𝑜 |
9 | pm54.43 9025 | . . . 4 ⊢ (((1𝑜 × {∅}) ≈ 1𝑜 ∧ (1𝑜 × {1𝑜}) ≈ 1𝑜) → (((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜)) | |
10 | 7, 8, 9 | mp2an 664 | . . 3 ⊢ (((1𝑜 × {∅}) ∩ (1𝑜 × {1𝑜})) = ∅ ↔ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜) |
11 | 4, 10 | mpbi 220 | . 2 ⊢ ((1𝑜 × {∅}) ∪ (1𝑜 × {1𝑜})) ≈ 2𝑜 |
12 | 3, 11 | eqbrtri 4807 | 1 ⊢ (1𝑜 +𝑐 1𝑜) ≈ 2𝑜 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1631 ∈ wcel 2145 ∪ cun 3721 ∩ cin 3722 ∅c0 4063 {csn 4316 class class class wbr 4786 × cxp 5247 Oncon0 5866 (class class class)co 6792 1𝑜c1o 7705 2𝑜c2o 7706 ≈ cen 8105 +𝑐 ccda 9190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1o 7712 df-2o 7713 df-er 7895 df-en 8109 df-dom 8110 df-sdom 8111 df-cda 9191 |
This theorem is referenced by: (None) |
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