MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  enp1ilem Structured version   Visualization version   GIF version

Theorem enp1ilem 8484
Description: Lemma for uses of enp1i 8485. (Contributed by Mario Carneiro, 5-Jan-2016.)
Hypothesis
Ref Expression
enp1ilem.1 𝑇 = ({𝑥} ∪ 𝑆)
Assertion
Ref Expression
enp1ilem (𝑥𝐴 → ((𝐴 ∖ {𝑥}) = 𝑆𝐴 = 𝑇))

Proof of Theorem enp1ilem
StepHypRef Expression
1 uneq1 3983 . . 3 ((𝐴 ∖ {𝑥}) = 𝑆 → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝑆 ∪ {𝑥}))
2 undif1 4267 . . 3 ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥})
3 uncom 3980 . . . 4 (𝑆 ∪ {𝑥}) = ({𝑥} ∪ 𝑆)
4 enp1ilem.1 . . . 4 𝑇 = ({𝑥} ∪ 𝑆)
53, 4eqtr4i 2805 . . 3 (𝑆 ∪ {𝑥}) = 𝑇
61, 2, 53eqtr3g 2837 . 2 ((𝐴 ∖ {𝑥}) = 𝑆 → (𝐴 ∪ {𝑥}) = 𝑇)
7 snssi 4572 . . . 4 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
8 ssequn2 4009 . . . 4 ({𝑥} ⊆ 𝐴 ↔ (𝐴 ∪ {𝑥}) = 𝐴)
97, 8sylib 210 . . 3 (𝑥𝐴 → (𝐴 ∪ {𝑥}) = 𝐴)
109eqeq1d 2780 . 2 (𝑥𝐴 → ((𝐴 ∪ {𝑥}) = 𝑇𝐴 = 𝑇))
116, 10syl5ib 236 1 (𝑥𝐴 → ((𝐴 ∖ {𝑥}) = 𝑆𝐴 = 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  cdif 3789  cun 3790  wss 3792  {csn 4398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-sn 4399
This theorem is referenced by:  en2  8486  en3  8487  en4  8488
  Copyright terms: Public domain W3C validator