Theorem enp1ilem 9230

Description: Lemma for uses of enp1i 9231. (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 4127	. . 3 ⊢ ((𝐴 ∖ {𝑥}) = 𝑆 → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝑆 ∪ {𝑥}))
2		undif1 4442	. . 3 ⊢ ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥})
3		uncom 4124	. . . 4 ⊢ (𝑆 ∪ {𝑥}) = ({𝑥} ∪ 𝑆)
4		enp1ilem.1	. . . 4 ⊢ 𝑇 = ({𝑥} ∪ 𝑆)
5	3, 4	eqtr4i 2756	. . 3 ⊢ (𝑆 ∪ {𝑥}) = 𝑇
6	1, 2, 5	3eqtr3g 2788	. 2 ⊢ ((𝐴 ∖ {𝑥}) = 𝑆 → (𝐴 ∪ {𝑥}) = 𝑇)
7		snssi 4775	. . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
8		ssequn2 4155	. . . 4 ⊢ ({𝑥} ⊆ 𝐴 ↔ (𝐴 ∪ {𝑥}) = 𝐴)
9	7, 8	sylib 218	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐴 ∪ {𝑥}) = 𝐴)
10	9	eqeq1d 2732	. 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 1540   ∈ wcel 2109   ∖ cdif 3914   ∪ cun 3915   ⊆ wss 3917  {csn 4592

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-sn 4593

This theorem is referenced by:  en2  9233  en3  9234  en4  9235