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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 3557 | . 2 ⊢ (∪ 𝐴 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ 𝐴)) | |
2 | df-tr 4681 | . 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
3 | 19.23v 1889 | . . . 4 ⊢ (∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)) | |
4 | eluni 4375 | . . . . 5 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) | |
5 | 4 | imbi1i 338 | . . . 4 ⊢ ((𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ 𝐴) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)) |
6 | 3, 5 | bitr4i 266 | . . 3 ⊢ (∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ 𝐴)) |
7 | 6 | albii 1737 | . 2 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ 𝐴)) |
8 | 1, 2, 7 | 3bitr4i 291 | 1 ⊢ (Tr 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 ∃wex 1695 ∈ wcel 1977 ⊆ wss 3540 ∪ cuni 4372 Tr wtr 4680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-uni 4373 df-tr 4681 |
This theorem is referenced by: dftr5 4683 trel 4687 ordelord 5662 suctr 5725 suctrOLD 5726 ordom 6966 hartogs 8332 card2on 8342 trcl 8487 tskwe 8659 ondomon 9264 dftr6 30893 elpotr 30930 hftr 31459 dford4 36614 tratrb 37767 trsbc 37771 truniALT 37772 sspwtr 38070 sspwtrALT 38071 sspwtrALT2 38080 pwtrVD 38081 pwtrrVD 38082 suctrALT 38083 suctrALT2VD 38093 suctrALT2 38094 tratrbVD 38119 trsbcVD 38135 truniALTVD 38136 trintALTVD 38138 trintALT 38139 suctrALTcf 38180 suctrALTcfVD 38181 suctrALT3 38182 |
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