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Mirrors > Home > MPE Home > Th. List > Mathboxes > trelded | Structured version Visualization version GIF version |
Description: Deduction form of trel 5235. In a transitive class, the membership relation is transitive. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
trelded.1 | ⊢ (𝜑 → Tr 𝐴) |
trelded.2 | ⊢ (𝜓 → 𝐵 ∈ 𝐶) |
trelded.3 | ⊢ (𝜒 → 𝐶 ∈ 𝐴) |
Ref | Expression |
---|---|
trelded | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝐵 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trelded.1 | . 2 ⊢ (𝜑 → Tr 𝐴) | |
2 | trelded.2 | . 2 ⊢ (𝜓 → 𝐵 ∈ 𝐶) | |
3 | trelded.3 | . 2 ⊢ (𝜒 → 𝐶 ∈ 𝐴) | |
4 | trel 5235 | . . 3 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴)) | |
5 | 4 | 3impib 1117 | . 2 ⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴) |
6 | 1, 2, 3, 5 | syl3an 1161 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 ∈ wcel 2107 Tr wtr 5226 |
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-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-3an 1090 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3449 df-in 3921 df-ss 3931 df-uni 4870 df-tr 5227 |
This theorem is referenced by: suctrALT3 43298 |
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