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Theorem enp1ilem 8730
Description: Lemma for uses of enp1i 8731. (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 4111 . . 3 ((𝐴 ∖ {𝑥}) = 𝑆 → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝑆 ∪ {𝑥}))
2 undif1 4400 . . 3 ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥})
3 uncom 4108 . . . 4 (𝑆 ∪ {𝑥}) = ({𝑥} ∪ 𝑆)
4 enp1ilem.1 . . . 4 𝑇 = ({𝑥} ∪ 𝑆)
53, 4eqtr4i 2846 . . 3 (𝑆 ∪ {𝑥}) = 𝑇
61, 2, 53eqtr3g 2878 . 2 ((𝐴 ∖ {𝑥}) = 𝑆 → (𝐴 ∪ {𝑥}) = 𝑇)
7 snssi 4717 . . . 4 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
8 ssequn2 4138 . . . 4 ({𝑥} ⊆ 𝐴 ↔ (𝐴 ∪ {𝑥}) = 𝐴)
97, 8sylib 220 . . 3 (𝑥𝐴 → (𝐴 ∪ {𝑥}) = 𝐴)
109eqeq1d 2822 . 2 (𝑥𝐴 → ((𝐴 ∪ {𝑥}) = 𝑇𝐴 = 𝑇))
116, 10syl5ib 246 1 (𝑥𝐴 → ((𝐴 ∖ {𝑥}) = 𝑆𝐴 = 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2114  cdif 3910  cun 3911  wss 3913  {csn 4543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-sn 4544
This theorem is referenced by:  en2  8732  en3  8733  en4  8734
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