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Mirrors > Home > MPE Home > Th. List > Mathboxes > trelpss | Structured version Visualization version GIF version |
Description: An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 5659, ax-reg 9583 is required for its proof. (Contributed by Andrew Salmon, 13-Nov-2011.) |
Ref | Expression |
---|---|
trelpss | ⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊊ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zfregfr 9596 | . . 3 ⊢ E Fr 𝐴 | |
2 | tz7.2 5659 | . . 3 ⊢ ((Tr 𝐴 ∧ E Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴)) | |
3 | 1, 2 | mp3an2 1449 | . 2 ⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴)) |
4 | df-pss 3966 | . 2 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴)) | |
5 | 3, 4 | sylibr 233 | 1 ⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊊ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ≠ wne 2940 ⊆ wss 3947 ⊊ wpss 3948 Tr wtr 5264 E cep 5578 Fr wfr 5627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-reg 9583 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5579 df-fr 5630 |
This theorem is referenced by: (None) |
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