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Theorem rintn0 4056
 Description: Relative intersection of a nonempty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)
Assertion
Ref Expression
rintn0 ((X A X) → (AX) = X)

Proof of Theorem rintn0
StepHypRef Expression
1 incom 3448 . 2 (AX) = (XA)
2 intssuni2 3951 . . . 4 ((X A X) → X A)
3 ssid 3290 . . . . 5 A A
4 sspwuni 4051 . . . . 5 (A AA A)
53, 4mpbi 199 . . . 4 A A
62, 5syl6ss 3284 . . 3 ((X A X) → X A)
7 df-ss 3259 . . 3 (X A ↔ (XA) = X)
86, 7sylib 188 . 2 ((X A X) → (XA) = X)
91, 8syl5eq 2397 1 ((X A X) → (AX) = X)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ≠ wne 2516   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550  ℘cpw 3722  ∪cuni 3891  ∩cint 3926 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-uni 3892  df-int 3927 This theorem is referenced by: (None)
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