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Mirrors > Home > MPE Home > Th. List > tz7.2 | Structured version Visualization version GIF version |
Description: Similar to Theorem 7.2 of [TakeutiZaring] p. 35, except that the Axiom of Regularity is not required due to the antecedent E Fr 𝐴. (Contributed by NM, 4-May-1994.) |
Ref | Expression |
---|---|
tz7.2 | ⊢ ((Tr 𝐴 ∧ E Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trss 5276 | . . 3 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) | |
2 | efrirr 5669 | . . . . 5 ⊢ ( E Fr 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
3 | eleq1 2827 | . . . . . 6 ⊢ (𝐵 = 𝐴 → (𝐵 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴)) | |
4 | 3 | notbid 318 | . . . . 5 ⊢ (𝐵 = 𝐴 → (¬ 𝐵 ∈ 𝐴 ↔ ¬ 𝐴 ∈ 𝐴)) |
5 | 2, 4 | syl5ibrcom 247 | . . . 4 ⊢ ( E Fr 𝐴 → (𝐵 = 𝐴 → ¬ 𝐵 ∈ 𝐴)) |
6 | 5 | necon2ad 2953 | . . 3 ⊢ ( E Fr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ≠ 𝐴)) |
7 | 1, 6 | anim12ii 618 | . 2 ⊢ ((Tr 𝐴 ∧ E Fr 𝐴) → (𝐵 ∈ 𝐴 → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴))) |
8 | 7 | 3impia 1116 | 1 ⊢ ((Tr 𝐴 ∧ E Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 Tr wtr 5265 E cep 5588 Fr wfr 5638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-fr 5641 |
This theorem is referenced by: tz7.7 6412 trelpss 44451 |
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