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Mirrors > Home > MPE Home > Th. List > tru | Structured version Visualization version GIF version |
Description: The truth value ⊤ is provable. (Contributed by Anthony Hart, 13-Oct-2010.) |
Ref | Expression |
---|---|
tru | ⊢ ⊤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥) | |
2 | df-tru 1539 | . 2 ⊢ (⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)) | |
3 | 1, 2 | mpbir 231 | 1 ⊢ ⊤ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1534 = wceq 1536 ⊤wtru 1537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 207 df-tru 1539 |
This theorem is referenced by: dftru2 1541 trut 1542 mptru 1543 tbtru 1544 bitru 1545 trud 1546 truan 1547 fal 1550 truorfal 1574 falortru 1575 cadtru 1616 nftru 1800 altru 1803 extru 1972 sbtru 2064 vextru 2718 rextru 3074 rabtru 3691 ss2abi 4076 ab0orv 4388 disjprg 5143 reusv2lem5 5407 rabxfr 5423 reuhyp 5425 euotd 5522 mptexgf 7241 elabrex 7261 elabrexg 7262 caovcl 7626 caovass 7632 caovdi 7651 ectocl 8823 fin1a2lem10 10446 riotaneg 12244 zriotaneg 12728 cshwsexaOLD 14859 eflt 16149 efgi0 19752 efgi1 19753 0frgp 19811 mpomulcn 24904 iundisj2 25597 pige3ALT 26576 tanord1 26593 tanord 26594 logtayl 26716 n0sind 28351 nnsind 28377 iundisj2f 32609 iundisj2fi 32804 ordtconn 33885 tgoldbachgt 34656 nexntru 36386 bj-fal 36552 bj-axd2d 36575 bj-rabtr 36912 bj-rabtrALT 36913 bj-dfid2ALT 37047 bj-finsumval0 37267 wl-impchain-mp-x 37429 wl-impchain-com-1.x 37433 wl-impchain-com-n.m 37438 wl-impchain-a1-x 37442 wl-moteq 37494 ftc1anclem5 37683 lhpexle1 39990 3lexlogpow5ineq2 42036 3lexlogpow2ineq1 42039 3lexlogpow2ineq2 42040 mzpcompact2lem 42738 ifpdfor 43454 ifpim1 43458 ifpnot 43459 ifpid2 43460 ifpim2 43461 uun0.1 44775 uunT1 44777 un10 44785 un01 44786 liminfvalxr 45738 ovn02 46523 rmotru 48651 reutru 48652 |
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