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Theorem enp1ilem 9312
Description: Lemma for uses of enp1i 9313. (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 4161 . . 3 ((𝐴 ∖ {𝑥}) = 𝑆 → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝑆 ∪ {𝑥}))
2 undif1 4476 . . 3 ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥})
3 uncom 4158 . . . 4 (𝑆 ∪ {𝑥}) = ({𝑥} ∪ 𝑆)
4 enp1ilem.1 . . . 4 𝑇 = ({𝑥} ∪ 𝑆)
53, 4eqtr4i 2768 . . 3 (𝑆 ∪ {𝑥}) = 𝑇
61, 2, 53eqtr3g 2800 . 2 ((𝐴 ∖ {𝑥}) = 𝑆 → (𝐴 ∪ {𝑥}) = 𝑇)
7 snssi 4808 . . . 4 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
8 ssequn2 4189 . . . 4 ({𝑥} ⊆ 𝐴 ↔ (𝐴 ∪ {𝑥}) = 𝐴)
97, 8sylib 218 . . 3 (𝑥𝐴 → (𝐴 ∪ {𝑥}) = 𝐴)
109eqeq1d 2739 . 2 (𝑥𝐴 → ((𝐴 ∪ {𝑥}) = 𝑇𝐴 = 𝑇))
116, 10imbitrid 244 1 (𝑥𝐴 → ((𝐴 ∖ {𝑥}) = 𝑆𝐴 = 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cdif 3948  cun 3949  wss 3951  {csn 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-sn 4627
This theorem is referenced by:  en2  9315  en3  9316  en4  9317
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