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Type | Label | Description |
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Statement | ||
Theorem | pm10.12 36701* | Theorem *10.12 in [WhiteheadRussell] p. 146. In *10, this is treated as an axiom, and the proofs in *10 are based on this theorem. (Contributed by Andrew Salmon, 17-Jun-2011.) |
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Theorem | pm10.14 36702 | Theorem *10.14 in [WhiteheadRussell] p. 146. (Contributed by Andrew Salmon, 17-Jun-2011.) |
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Theorem | pm10.251 36703 | Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) |
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Theorem | pm10.252 36704 | Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) (New usage is discouraged.) |
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Theorem | pm10.253 36705 | Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) |
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Theorem | albitr 36706 | Theorem *10.301 in [WhiteheadRussell] p. 151. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | pm10.42 36707 | Theorem *10.42 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 17-Jun-2011.) |
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Theorem | pm10.52 36708* | Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | pm10.53 36709 | Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | pm10.541 36710* | Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | pm10.542 36711* | Theorem *10.542 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | pm10.55 36712 | Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | pm10.56 36713 | Theorem *10.56 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | pm10.57 36714 | Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | 2alanimi 36715 | Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | 2al2imi 36716 | Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | pm11.11 36717 | Theorem *11.11 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.) |
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Theorem | pm11.12 36718* | Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.) |
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Theorem | 19.21vv 36719* | Compare Theorem *11.3 in [WhiteheadRussell] p. 161. Special case of theorem 19.21 of [Margaris] p. 90 with two quantifiers. See 19.21v 1785. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | 2alim 36720 | Theorem *11.32 in [WhiteheadRussell] p. 162. Theorem 19.20 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | 2albi 36721 | Theorem *11.33 in [WhiteheadRussell] p. 162. Theorem 19.15 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | 2exim 36722 | Theorem *11.34 in [WhiteheadRussell] p. 162. Theorem 19.22 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | 2exbi 36723 | Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | spsbce-2 36724 | Theorem *11.36 in [WhiteheadRussell] p. 162. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | 19.33-2 36725 | Theorem *11.421 in [WhiteheadRussell] p. 163. Theorem 19.33 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | 19.36vv 36726* | Theorem *11.43 in [WhiteheadRussell] p. 163. Theorem 19.36 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011.) |
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Theorem | 19.31vv 36727* | Theorem *11.44 in [WhiteheadRussell] p. 163. Theorem 19.31 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | 19.37vv 36728* | Theorem *11.46 in [WhiteheadRussell] p. 164. Theorem 19.37 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | 19.28vv 36729* | Theorem *11.47 in [WhiteheadRussell] p. 164. Theorem 19.28 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | pm11.52 36730 | Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | 2exanali 36731 | Theorem *11.521 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | aaanv 36732* | Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2054. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | pm11.57 36733* | Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | pm11.58 36734* | Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | pm11.59 36735* | Theorem *11.59 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.) |
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Theorem | pm11.6 36736* | Theorem *11.6 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.) |
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Theorem | pm11.61 36737* | Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | pm11.62 36738* | Theorem *11.62 in [WhiteheadRussell] p. 166. Importation combined with the rearrangement with quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | pm11.63 36739 | Theorem *11.63 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | pm11.7 36740 | Theorem *11.7 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | pm11.71 36741* | Theorem *11.71 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | sbeqal1 36742* |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | sbeqal1i 36743* |
Suppose you know ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | sbeqal2i 36744* |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | sbeqalbi 36745* |
When both ![]() ![]() ![]() ![]() |
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Theorem | axc5c4c711 36746 | Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1668 as the inference rule. This proof extends the idea of axc5c711 32483 and related theorems. (Contributed by Andrew Salmon, 14-Jul-2011.) |
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Theorem | axc5c4c711toc5 36747 | Re-derivation of sp 1936 from axc5c4c711 36746. Note that ax6 2094 is used for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) Revised to use ax6v 1805 instead of ax6 2094, so that this re-derivation requires only ax6v 1805 and propositional calculus. (Revised by BJ, 14-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | axc5c4c711toc4 36748 | Re-derivation of axc4 1937 from axc5c4c711 36746. Note that only propositional calculus is required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | axc5c4c711toc7 36749 | Re-derivation of axc7 1938 from axc5c4c711 36746. Note that neither axc7 1938 nor ax-11 1919 are required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | axc5c4c711to11 36750 | Re-derivation of ax-11 1919 from axc5c4c711 36746. Note that ax-11 1919 is not required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | axc11next 36751* | This theorem shows that, given axext4 2434, we can derive a version of axc11n 2142. However, it is weaker than axc11n 2142 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 16-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | pm13.13a 36752 | One result of theorem *13.13 in [WhiteheadRussell] p. 178. A note on the section - to make the theorems more usable, and because inequality is notation for set theory (it is not defined in the predicate calculus section), this section will use classes instead of sets. (Contributed by Andrew Salmon, 3-Jun-2011.) |
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Theorem | pm13.13b 36753 | Theorem *13.13 in [WhiteheadRussell] p. 178 with different variable substitution. (Contributed by Andrew Salmon, 3-Jun-2011.) |
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Theorem | pm13.14 36754 | Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.) |
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Theorem | pm13.192 36755* | Theorem *13.192 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.) |
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Theorem | pm13.193 36756 | Theorem *13.193 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) |
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Theorem | pm13.194 36757 | Theorem *13.194 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) |
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Theorem | pm13.195 36758* | Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3291. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.) |
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Theorem | pm13.196a 36759* | Theorem *13.196 in [WhiteheadRussell] p. 179. The only difference is the position of the substituted variable. (Contributed by Andrew Salmon, 3-Jun-2011.) |
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Theorem | 2sbc6g 36760* | Theorem *13.21 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) |
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Theorem | 2sbc5g 36761* | Theorem *13.22 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) |
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Theorem | iotain 36762 |
Equivalence between two different forms of ![]() |
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Theorem | iotaexeu 36763 | The iota class exists. This theorem does not require ax-nul 4533 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.) |
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Theorem | iotasbc 36764* |
Definition *14.01 in [WhiteheadRussell] p. 184. In Principia
Mathematica, Russell and Whitehead define ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | iotasbc2 36765* | Theorem *14.111 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.) |
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Theorem | pm14.12 36766* | Theorem *14.12 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.) |
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Theorem | pm14.122a 36767* | Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
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Theorem | pm14.122b 36768* | Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
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Theorem | pm14.122c 36769* | Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
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Theorem | pm14.123a 36770* | Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
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Theorem | pm14.123b 36771* | Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
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Theorem | pm14.123c 36772* | Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.) |
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Theorem | pm14.18 36773 | Theorem *14.18 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.) |
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Theorem | iotaequ 36774* | Theorem *14.2 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.) |
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Theorem | iotavalb 36775* | Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 5556. (Contributed by Andrew Salmon, 11-Jul-2011.) |
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Theorem | iotasbc5 36776* | Theorem *14.205 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 11-Jul-2011.) |
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Theorem | pm14.24 36777* | Theorem *14.24 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.) |
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Theorem | iotavalsb 36778* | Theorem *14.242 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) |
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Theorem | sbiota1 36779 | Theorem *14.25 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.) |
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Theorem | sbaniota 36780 | Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.) |
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Theorem | eubi 36781 | Theorem *14.271 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) |
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Theorem | iotasbcq 36782 | Theorem *14.272 in [WhiteheadRussell] p. 193. (Contributed by Andrew Salmon, 11-Jul-2011.) |
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Theorem | elnev 36783* | Any set that contains one element less than the universe is not equal to it. (Contributed by Andrew Salmon, 16-Jun-2011.) |
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Theorem | rusbcALT 36784 | A version of Russell's paradox which is proven using proper substitution. (Contributed by Andrew Salmon, 18-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | compel 36785 |
Equivalence between two ways of saying "is a member of the complement of
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Theorem | compeq 36786* |
Equality between two ways of saying "the complement of ![]() |
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Theorem | compne 36787 |
The complement of ![]() ![]() |
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Theorem | compab 36788 | Two ways of saying "the complement of a class abstraction". (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
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Theorem | conss34 36789 | Contrpositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.) |
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Theorem | conss2 36790 | Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.) |
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Theorem | conss1 36791 | Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.) |
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Theorem | ralbidar 36792 | More general form of ralbida 2820. (Contributed by Andrew Salmon, 25-Jul-2011.) |
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Theorem | rexbidar 36793 | More general form of rexbida 2895. (Contributed by Andrew Salmon, 25-Jul-2011.) |
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Theorem | dropab1 36794 | Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.) |
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Theorem | dropab2 36795 | Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.) |
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Theorem | ipo0 36796 | If the identity relation partially orders any class, then that class is the null class. (Contributed by Andrew Salmon, 25-Jul-2011.) |
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Theorem | ifr0 36797 | A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011.) |
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Theorem | ordpss 36798 | ordelpss 5450 with an antecedent removed. (Contributed by Andrew Salmon, 25-Jul-2011.) |
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Theorem | fvsb 36799* | Explicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.) |
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Theorem | fveqsb 36800* | Implicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.) |
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