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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 3901 | . 2 ⊢ (∪ 𝐴 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ 𝐴)) | |
2 | df-tr 5137 | . 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 4803 | . . . . 5 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) | |
5 | 4 | imbi1i 353 | . . . 4 ⊢ ((𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ 𝐴) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)) |
6 | 3, 5 | bitr4i 281 | . . 3 ⊢ (∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ 𝐴)) |
7 | 6 | albii 1821 | . 2 ⊢ (∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝑥 ∈ ∪ 𝐴 → 𝑥 ∈ 𝐴)) |
8 | 1, 2, 7 | 3bitr4i 306 | 1 ⊢ (Tr 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 ∃wex 1781 ∈ wcel 2111 ⊆ wss 3881 ∪ cuni 4800 Tr wtr 5136 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 df-uni 4801 df-tr 5137 |
This theorem is referenced by: dftr5 5139 trel 5143 ordelord 6181 suctr 6242 ordom 7569 hartogs 8992 card2on 9002 trcl 9154 tskwe 9363 ondomon 9974 dftr6 33099 elpotr 33139 nosupno 33316 hftr 33756 dford4 39970 mnutrd 40988 tratrb 41242 trsbc 41246 truniALT 41247 sspwtr 41527 sspwtrALT 41528 sspwtrALT2 41529 pwtrVD 41530 pwtrrVD 41531 suctrALT 41532 suctrALT2VD 41542 suctrALT2 41543 tratrbVD 41567 trsbcVD 41583 truniALTVD 41584 trintALTVD 41586 trintALT 41587 suctrALTcf 41628 suctrALTcfVD 41629 suctrALT3 41630 |
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