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Theorem enp1ilem 8154
Description: Lemma for uses of enp1i 8155. (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 3744 . . 3 ((𝐴 ∖ {𝑥}) = 𝑆 → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝑆 ∪ {𝑥}))
2 undif1 4021 . . 3 ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥})
3 uncom 3741 . . . 4 (𝑆 ∪ {𝑥}) = ({𝑥} ∪ 𝑆)
4 enp1ilem.1 . . . 4 𝑇 = ({𝑥} ∪ 𝑆)
53, 4eqtr4i 2646 . . 3 (𝑆 ∪ {𝑥}) = 𝑇
61, 2, 53eqtr3g 2678 . 2 ((𝐴 ∖ {𝑥}) = 𝑆 → (𝐴 ∪ {𝑥}) = 𝑇)
7 snssi 4315 . . . 4 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
8 ssequn2 3770 . . . 4 ({𝑥} ⊆ 𝐴 ↔ (𝐴 ∪ {𝑥}) = 𝐴)
97, 8sylib 208 . . 3 (𝑥𝐴 → (𝐴 ∪ {𝑥}) = 𝐴)
109eqeq1d 2623 . 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 1480  wcel 1987  cdif 3557  cun 3558  wss 3560  {csn 4155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-sn 4156
This theorem is referenced by:  en2  8156  en3  8157  en4  8158
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