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Mirrors > Home > MPE Home > Th. List > Mathboxes > trelded | Structured version Visualization version GIF version |
Description: Deduction form of trel 5185. 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 5185 | . . 3 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴)) | |
5 | 4 | 3impib 1118 | . 2 ⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴) |
6 | 1, 2, 3, 5 | syl3an 1162 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 ∈ wcel 2112 Tr wtr 5178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1091 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-v 3425 df-in 3890 df-ss 3900 df-uni 4837 df-tr 5179 |
This theorem is referenced by: suctrALT3 42265 |
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