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Mirrors > Home > ILE Home > Th. List > trss | GIF version |
Description: An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.) |
Ref | Expression |
---|---|
trss | ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2180 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
2 | sseq1 3090 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴)) | |
3 | 1, 2 | imbi12d 233 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴) ↔ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))) |
4 | 3 | imbi2d 229 | . . 3 ⊢ (𝑥 = 𝐵 → ((Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) ↔ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)))) |
5 | dftr3 4000 | . . . 4 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴) | |
6 | rsp 2457 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) | |
7 | 5, 6 | sylbi 120 | . . 3 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) |
8 | 4, 7 | vtoclg 2720 | . 2 ⊢ (𝐵 ∈ 𝐴 → (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))) |
9 | 8 | pm2.43b 52 | 1 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 ∀wral 2393 ⊆ wss 3041 Tr wtr 3996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-in 3047 df-ss 3054 df-uni 3707 df-tr 3997 |
This theorem is referenced by: trin 4006 triun 4009 trintssm 4012 tz7.2 4246 ordelss 4271 trsucss 4315 ordsucss 4390 ctinf 11870 |
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