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Theorem enp1ilem 8361
Description: Lemma for uses of enp1i 8362. (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 3903 . . 3 ((𝐴 ∖ {𝑥}) = 𝑆 → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝑆 ∪ {𝑥}))
2 undif1 4187 . . 3 ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥})
3 uncom 3900 . . . 4 (𝑆 ∪ {𝑥}) = ({𝑥} ∪ 𝑆)
4 enp1ilem.1 . . . 4 𝑇 = ({𝑥} ∪ 𝑆)
53, 4eqtr4i 2785 . . 3 (𝑆 ∪ {𝑥}) = 𝑇
61, 2, 53eqtr3g 2817 . 2 ((𝐴 ∖ {𝑥}) = 𝑆 → (𝐴 ∪ {𝑥}) = 𝑇)
7 snssi 4484 . . . 4 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
8 ssequn2 3929 . . . 4 ({𝑥} ⊆ 𝐴 ↔ (𝐴 ∪ {𝑥}) = 𝐴)
97, 8sylib 208 . . 3 (𝑥𝐴 → (𝐴 ∪ {𝑥}) = 𝐴)
109eqeq1d 2762 . 2 (𝑥𝐴 → ((𝐴 ∪ {𝑥}) = 𝑇𝐴 = 𝑇))
116, 10syl5ib 234 1 (𝑥𝐴 → ((𝐴 ∖ {𝑥}) = 𝑆𝐴 = 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  cdif 3712  cun 3713  wss 3715  {csn 4321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-sn 4322
This theorem is referenced by:  en2  8363  en3  8364  en4  8365
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