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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | bnj534 32301* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜒 → (∃𝑥𝜑 ∧ 𝜓)) ⇒ ⊢ (𝜒 → ∃𝑥(𝜑 ∧ 𝜓)) | ||
Theorem | bnj538 32302* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) (Proof shortened by OpenAI, 30-Mar-2020.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ([𝐴 / 𝑦]∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐵 [𝐴 / 𝑦]𝜑) | ||
Theorem | bnj529 32303 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 𝐷 = (ω ∖ {∅}) ⇒ ⊢ (𝑀 ∈ 𝐷 → ∅ ∈ 𝑀) | ||
Theorem | bnj551 32304 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝑚 = suc 𝑝 ∧ 𝑚 = suc 𝑖) → 𝑝 = 𝑖) | ||
Theorem | bnj563 32305 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)) & ⊢ (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖)) ⇒ ⊢ ((𝜂 ∧ 𝜌) → suc 𝑖 ∈ 𝑚) | ||
Theorem | bnj564 32306 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)) ⇒ ⊢ (𝜏 → dom 𝑓 = 𝑚) | ||
Theorem | bnj593 32307 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∃𝑥𝜓) & ⊢ (𝜓 → 𝜒) ⇒ ⊢ (𝜑 → ∃𝑥𝜒) | ||
Theorem | bnj596 32308 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → ∃𝑥𝜓) ⇒ ⊢ (𝜑 → ∃𝑥(𝜑 ∧ 𝜓)) | ||
Theorem | bnj610 32309* | Pass from equality (𝑥 = 𝐴) to substitution ([𝐴 / 𝑥]) without the distinct variable condition on 𝐴, 𝑥. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓′)) & ⊢ (𝑦 = 𝐴 → (𝜓′ ↔ 𝜓)) ⇒ ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) | ||
Theorem | bnj642 32310 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜑) | ||
Theorem | bnj643 32311 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜓) | ||
Theorem | bnj645 32312 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜃) | ||
Theorem | bnj658 32313 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → (𝜑 ∧ 𝜓 ∧ 𝜒)) | ||
Theorem | bnj667 32314 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → (𝜓 ∧ 𝜒 ∧ 𝜃)) | ||
Theorem | bnj705 32315 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜏) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) | ||
Theorem | bnj706 32316 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜓 → 𝜏) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) | ||
Theorem | bnj707 32317 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜒 → 𝜏) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) | ||
Theorem | bnj708 32318 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜃 → 𝜏) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) | ||
Theorem | bnj721 32319 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜏) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) | ||
Theorem | bnj832 32320 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓)) & ⊢ (𝜑 → 𝜏) ⇒ ⊢ (𝜂 → 𝜏) | ||
Theorem | bnj835 32321 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒)) & ⊢ (𝜑 → 𝜏) ⇒ ⊢ (𝜂 → 𝜏) | ||
Theorem | bnj836 32322 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒)) & ⊢ (𝜓 → 𝜏) ⇒ ⊢ (𝜂 → 𝜏) | ||
Theorem | bnj837 32323 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒)) & ⊢ (𝜒 → 𝜏) ⇒ ⊢ (𝜂 → 𝜏) | ||
Theorem | bnj769 32324 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃)) & ⊢ (𝜑 → 𝜏) ⇒ ⊢ (𝜂 → 𝜏) | ||
Theorem | bnj770 32325 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃)) & ⊢ (𝜓 → 𝜏) ⇒ ⊢ (𝜂 → 𝜏) | ||
Theorem | bnj771 32326 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜂 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃)) & ⊢ (𝜒 → 𝜏) ⇒ ⊢ (𝜂 → 𝜏) | ||
Theorem | bnj887 32327 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 ↔ 𝜑′) & ⊢ (𝜓 ↔ 𝜓′) & ⊢ (𝜒 ↔ 𝜒′) & ⊢ (𝜃 ↔ 𝜃′) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝜒′ ∧ 𝜃′)) | ||
Theorem | bnj918 32328 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) ⇒ ⊢ 𝐺 ∈ V | ||
Theorem | bnj919 32329* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝐹 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) & ⊢ (𝜑′ ↔ [𝑃 / 𝑛]𝜑) & ⊢ (𝜓′ ↔ [𝑃 / 𝑛]𝜓) & ⊢ (𝜒′ ↔ [𝑃 / 𝑛]𝜒) & ⊢ 𝑃 ∈ V ⇒ ⊢ (𝜒′ ↔ (𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′)) | ||
Theorem | bnj923 32330 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 𝐷 = (ω ∖ {∅}) ⇒ ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω) | ||
Theorem | bnj927 32331 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) & ⊢ 𝐶 ∈ V ⇒ ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝) | ||
Theorem | bnj930 32332 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐹 Fn 𝐴) ⇒ ⊢ (𝜑 → Fun 𝐹) | ||
Theorem | bnj931 32333 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 𝐴 = (𝐵 ∪ 𝐶) ⇒ ⊢ 𝐵 ⊆ 𝐴 | ||
Theorem | bnj937 32334* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∃𝑥𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | bnj941 32335 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) ⇒ ⊢ (𝐶 ∈ V → ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)) | ||
Theorem | bnj945 32336 | Technical lemma for bnj69 32573. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) ⇒ ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝐴 ∈ 𝑛) → (𝐺‘𝐴) = (𝑓‘𝐴)) | ||
Theorem | bnj946 32337 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ (𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | ||
Theorem | bnj951 32338 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜏 → 𝜑) & ⊢ (𝜏 → 𝜓) & ⊢ (𝜏 → 𝜒) & ⊢ (𝜏 → 𝜃) ⇒ ⊢ (𝜏 → (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃)) | ||
Theorem | bnj956 32339 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝐴 = 𝐵 → ∀𝑥 𝐴 = 𝐵) ⇒ ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) | ||
Theorem | bnj976 32340* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜒 ↔ (𝑁 ∈ 𝐷 ∧ 𝑓 Fn 𝑁 ∧ 𝜑 ∧ 𝜓)) & ⊢ (𝜑′ ↔ [𝐺 / 𝑓]𝜑) & ⊢ (𝜓′ ↔ [𝐺 / 𝑓]𝜓) & ⊢ (𝜒′ ↔ [𝐺 / 𝑓]𝜒) & ⊢ 𝐺 ∈ V ⇒ ⊢ (𝜒′ ↔ (𝑁 ∈ 𝐷 ∧ 𝐺 Fn 𝑁 ∧ 𝜑′ ∧ 𝜓′)) | ||
Theorem | bnj982 32341 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝜒 → ∀𝑥𝜒) & ⊢ (𝜃 → ∀𝑥𝜃) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃)) | ||
Theorem | bnj1019 32342* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (∃𝑝(𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ↔ (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏)) | ||
Theorem | bnj1023 32343 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ∃𝑥(𝜑 → 𝜓) & ⊢ (𝜓 → 𝜒) ⇒ ⊢ ∃𝑥(𝜑 → 𝜒) | ||
Theorem | bnj1095 32344 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ (𝜑 → ∀𝑥𝜑) | ||
Theorem | bnj1096 32345* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏 ∧ 𝜑)) ⇒ ⊢ (𝜓 → ∀𝑥𝜓) | ||
Theorem | bnj1098 32346* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 𝐷 = (ω ∖ {∅}) ⇒ ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) | ||
Theorem | bnj1101 32347 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ∃𝑥(𝜑 → 𝜓) & ⊢ (𝜒 → 𝜑) ⇒ ⊢ ∃𝑥(𝜒 → 𝜓) | ||
Theorem | bnj1113 32348* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝐴 = 𝐵 → 𝐶 = 𝐷) ⇒ ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐶 𝐸 = ∪ 𝑥 ∈ 𝐷 𝐸) | ||
Theorem | bnj1109 32349 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ∃𝑥((𝐴 ≠ 𝐵 ∧ 𝜑) → 𝜓) & ⊢ ((𝐴 = 𝐵 ∧ 𝜑) → 𝜓) ⇒ ⊢ ∃𝑥(𝜑 → 𝜓) | ||
Theorem | bnj1131 32350 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ ∃𝑥𝜑 ⇒ ⊢ 𝜑 | ||
Theorem | bnj1138 32351 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 𝐴 = (𝐵 ∪ 𝐶) ⇒ ⊢ (𝑋 ∈ 𝐴 ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶)) | ||
Theorem | bnj1142 32352 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) | ||
Theorem | bnj1143 32353* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵 | ||
Theorem | bnj1146 32354* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵 | ||
Theorem | bnj1149 32355 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐵 ∈ V) ⇒ ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V) | ||
Theorem | bnj1185 32356* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∃𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥) | ||
Theorem | bnj1196 32357 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | ||
Theorem | bnj1198 32358 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∃𝑥𝜓) & ⊢ (𝜓′ ↔ 𝜓) ⇒ ⊢ (𝜑 → ∃𝑥𝜓′) | ||
Theorem | bnj1209 32359* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜒 → ∃𝑥 ∈ 𝐵 𝜑) & ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑)) ⇒ ⊢ (𝜒 → ∃𝑥𝜃) | ||
Theorem | bnj1211 32360 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | ||
Theorem | bnj1213 32361 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 𝐴 ⊆ 𝐵 & ⊢ (𝜃 → 𝑥 ∈ 𝐴) ⇒ ⊢ (𝜃 → 𝑥 ∈ 𝐵) | ||
Theorem | bnj1212 32362* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑} & ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜏)) ⇒ ⊢ (𝜃 → 𝑥 ∈ 𝐴) | ||
Theorem | bnj1219 32363 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜒 ↔ (𝜑 ∧ 𝜓 ∧ 𝜁)) & ⊢ (𝜃 ↔ (𝜒 ∧ 𝜏 ∧ 𝜂)) ⇒ ⊢ (𝜃 → 𝜓) | ||
Theorem | bnj1224 32364 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ¬ (𝜃 ∧ 𝜏 ∧ 𝜂) ⇒ ⊢ ((𝜃 ∧ 𝜏) → ¬ 𝜂) | ||
Theorem | bnj1230 32365* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑} ⇒ ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) | ||
Theorem | bnj1232 32366 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃 ∧ 𝜏)) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | bnj1235 32367 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃 ∧ 𝜏)) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | bnj1239 32368 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒) → ∃𝑥 ∈ 𝐴 𝜓) | ||
Theorem | bnj1238 32369 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) | ||
Theorem | bnj1241 32370 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜓 → 𝐶 = 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ⊆ 𝐵) | ||
Theorem | bnj1247 32371 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃 ∧ 𝜏)) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | bnj1254 32372 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃 ∧ 𝜏)) ⇒ ⊢ (𝜑 → 𝜏) | ||
Theorem | bnj1262 32373 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 𝐴 ⊆ 𝐵 & ⊢ (𝜑 → 𝐶 = 𝐴) ⇒ ⊢ (𝜑 → 𝐶 ⊆ 𝐵) | ||
Theorem | bnj1266 32374 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜒 → ∃𝑥(𝜑 ∧ 𝜓)) ⇒ ⊢ (𝜒 → ∃𝑥𝜓) | ||
Theorem | bnj1265 32375* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | bnj1275 32376 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∃𝑥(𝜓 ∧ 𝜒)) & ⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (𝜑 → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒)) | ||
Theorem | bnj1276 32377 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝜒 → ∀𝑥𝜒) & ⊢ (𝜃 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜃 → ∀𝑥𝜃) | ||
Theorem | bnj1292 32378 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 𝐴 = (𝐵 ∩ 𝐶) ⇒ ⊢ 𝐴 ⊆ 𝐵 | ||
Theorem | bnj1293 32379 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 𝐴 = (𝐵 ∩ 𝐶) ⇒ ⊢ 𝐴 ⊆ 𝐶 | ||
Theorem | bnj1294 32380 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) & ⊢ (𝜑 → 𝑥 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | bnj1299 32381 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) | ||
Theorem | bnj1304 32382 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∃𝑥𝜓) & ⊢ (𝜓 → 𝜒) & ⊢ (𝜓 → ¬ 𝜒) ⇒ ⊢ ¬ 𝜑 | ||
Theorem | bnj1316 32383* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) & ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) ⇒ ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) | ||
Theorem | bnj1317 32384* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 𝐴 = {𝑥 ∣ 𝜑} ⇒ ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | ||
Theorem | bnj1322 32385 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐴) | ||
Theorem | bnj1340 32386 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜓 → ∃𝑥𝜃) & ⊢ (𝜒 ↔ (𝜓 ∧ 𝜃)) & ⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ (𝜓 → ∃𝑥𝜒) | ||
Theorem | bnj1345 32387 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∃𝑥(𝜓 ∧ 𝜒)) & ⊢ (𝜃 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒)) & ⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (𝜑 → ∃𝑥𝜃) | ||
Theorem | bnj1350 32388* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜒 → ∀𝑥𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒)) | ||
Theorem | bnj1351 32389* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓)) | ||
Theorem | bnj1352 32390* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓)) | ||
Theorem | bnj1361 32391* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | ||
Theorem | bnj1366 32392* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.) |
⊢ (𝜓 ↔ (𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ∧ 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑})) ⇒ ⊢ (𝜓 → 𝐵 ∈ V) | ||
Theorem | bnj1379 32393* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 ↔ ∀𝑓 ∈ 𝐴 Fun 𝑓) & ⊢ 𝐷 = (dom 𝑓 ∩ dom 𝑔) & ⊢ (𝜓 ↔ (𝜑 ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷))) & ⊢ (𝜒 ↔ (𝜓 ∧ 〈𝑥, 𝑦〉 ∈ ∪ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ ∪ 𝐴)) & ⊢ (𝜃 ↔ (𝜒 ∧ 𝑓 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝑓)) & ⊢ (𝜏 ↔ (𝜃 ∧ 𝑔 ∈ 𝐴 ∧ 〈𝑥, 𝑧〉 ∈ 𝑔)) ⇒ ⊢ (𝜓 → Fun ∪ 𝐴) | ||
Theorem | bnj1383 32394* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 ↔ ∀𝑓 ∈ 𝐴 Fun 𝑓) & ⊢ 𝐷 = (dom 𝑓 ∩ dom 𝑔) & ⊢ (𝜓 ↔ (𝜑 ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷))) ⇒ ⊢ (𝜓 → Fun ∪ 𝐴) | ||
Theorem | bnj1385 32395* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 ↔ ∀𝑓 ∈ 𝐴 Fun 𝑓) & ⊢ 𝐷 = (dom 𝑓 ∩ dom 𝑔) & ⊢ (𝜓 ↔ (𝜑 ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷))) & ⊢ (𝑥 ∈ 𝐴 → ∀𝑓 𝑥 ∈ 𝐴) & ⊢ (𝜑′ ↔ ∀ℎ ∈ 𝐴 Fun ℎ) & ⊢ 𝐸 = (dom ℎ ∩ dom 𝑔) & ⊢ (𝜓′ ↔ (𝜑′ ∧ ∀ℎ ∈ 𝐴 ∀𝑔 ∈ 𝐴 (ℎ ↾ 𝐸) = (𝑔 ↾ 𝐸))) ⇒ ⊢ (𝜓 → Fun ∪ 𝐴) | ||
Theorem | bnj1386 32396* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 ↔ ∀𝑓 ∈ 𝐴 Fun 𝑓) & ⊢ 𝐷 = (dom 𝑓 ∩ dom 𝑔) & ⊢ (𝜓 ↔ (𝜑 ∧ ∀𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 (𝑓 ↾ 𝐷) = (𝑔 ↾ 𝐷))) & ⊢ (𝑥 ∈ 𝐴 → ∀𝑓 𝑥 ∈ 𝐴) ⇒ ⊢ (𝜓 → Fun ∪ 𝐴) | ||
Theorem | bnj1397 32397 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∃𝑥𝜓) & ⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | bnj1400 32398* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) ⇒ ⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥 | ||
Theorem | bnj1405 32399* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → 𝑋 ∈ ∪ 𝑦 ∈ 𝐴 𝐵) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝑋 ∈ 𝐵) | ||
Theorem | bnj1422 32400 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → Fun 𝐴) & ⊢ (𝜑 → dom 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 Fn 𝐵) |
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