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Mirrors > Home > MPE Home > Th. List > dftr2 | Structured version Visualization version GIF version |
Description: An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. (Contributed by NM, 24-Apr-1994.) |
Ref | Expression |
---|---|
dftr2 | ⊢ (Tr 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 3957 | . 2 ⊢ (∪ 𝐴 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ 𝐴)) | |
2 | df-tr 5175 | . 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
3 | 19.23v 1943 | . . . 4 ⊢ (∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)) | |
4 | eluni 4843 | . . . . 5 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) | |
5 | 4 | imbi1i 352 | . . . 4 ⊢ ((𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ 𝐴) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)) |
6 | 3, 5 | bitr4i 280 | . . 3 ⊢ (∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ 𝐴)) |
7 | 6 | albii 1820 | . 2 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ 𝐴)) |
8 | 1, 2, 7 | 3bitr4i 305 | 1 ⊢ (Tr 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 ∃wex 1780 ∈ wcel 2114 ⊆ wss 3938 ∪ cuni 4840 Tr wtr 5174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-in 3945 df-ss 3954 df-uni 4841 df-tr 5175 |
This theorem is referenced by: dftr5 5177 trel 5181 ordelord 6215 suctr 6276 ordom 7591 hartogs 9010 card2on 9020 trcl 9172 tskwe 9381 ondomon 9987 dftr6 32988 elpotr 33028 nosupno 33205 hftr 33645 dford4 39633 mnutrd 40623 tratrb 40877 trsbc 40881 truniALT 40882 sspwtr 41162 sspwtrALT 41163 sspwtrALT2 41164 pwtrVD 41165 pwtrrVD 41166 suctrALT 41167 suctrALT2VD 41177 suctrALT2 41178 tratrbVD 41202 trsbcVD 41218 truniALTVD 41219 trintALTVD 41221 trintALT 41222 suctrALTcf 41263 suctrALTcfVD 41264 suctrALT3 41265 |
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