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Mirrors > Home > MPE Home > Th. List > Mathboxes > trelded | Structured version Visualization version GIF version |
Description: Deduction form of trel 4984. 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 4984 | . . 3 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴)) | |
5 | 4 | 3impib 1148 | . 2 ⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴) |
6 | 1, 2, 3, 5 | syl3an 1203 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1111 ∈ wcel 2164 Tr wtr 4977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-v 3416 df-in 3805 df-ss 3812 df-uni 4661 df-tr 4978 |
This theorem is referenced by: suctrALT3 39973 |
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