Theorem enp1ilem 9190

Description: Lemma for uses of enp1i 9191. (Contributed by Mario Carneiro, 5-Jan-2016.)

Hypothesis

Ref	Expression
enp1ilem.1	⊢ 𝑇 = ({𝑥} ∪ 𝑆)

Assertion

Ref	Expression
enp1ilem	⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∖ {𝑥}) = 𝑆 → 𝐴 = 𝑇))

Proof of Theorem enp1ilem

Step	Hyp	Ref	Expression
1		uneq1 4115	. . 3 ⊢ ((𝐴 ∖ {𝑥}) = 𝑆 → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝑆 ∪ {𝑥}))
2		undif1 4430	. . 3 ⊢ ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥})
3		uncom 4112	. . . 4 ⊢ (𝑆 ∪ {𝑥}) = ({𝑥} ∪ 𝑆)
4		enp1ilem.1	. . . 4 ⊢ 𝑇 = ({𝑥} ∪ 𝑆)
5	3, 4	eqtr4i 2763	. . 3 ⊢ (𝑆 ∪ {𝑥}) = 𝑇
6	1, 2, 5	3eqtr3g 2795	. 2 ⊢ ((𝐴 ∖ {𝑥}) = 𝑆 → (𝐴 ∪ {𝑥}) = 𝑇)
7		snssi 4766	. . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
8		ssequn2 4143	. . . 4 ⊢ ({𝑥} ⊆ 𝐴 ↔ (𝐴 ∪ {𝑥}) = 𝐴)
9	7, 8	sylib 218	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐴 ∪ {𝑥}) = 𝐴)
10	9	eqeq1d 2739	. 2 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∪ {𝑥}) = 𝑇 ↔ 𝐴 = 𝑇))
11	6, 10	imbitrid 244	1 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∖ {𝑥}) = 𝑆 → 𝐴 = 𝑇))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∖ cdif 3900   ∪ cun 3901   ⊆ wss 3903  {csn 4582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-sn 4583

This theorem is referenced by:  en2  9192  en3  9193  en4  9194