Theorem enp1ilem 9277

Description: Lemma for uses of enp1i 9278. (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 4159	. . 3 ⊢ ((𝐴 ∖ {𝑥}) = 𝑆 → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝑆 ∪ {𝑥}))
2		undif1 4429	. . 3 ⊢ ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥})
3		uncom 4156	. . . 4 ⊢ (𝑆 ∪ {𝑥}) = ({𝑥} ∪ 𝑆)
4		enp1ilem.1	. . . 4 ⊢ 𝑇 = ({𝑥} ∪ 𝑆)
5	3, 4	eqtr4i 2786	. . 3 ⊢ (𝑆 ∪ {𝑥}) = 𝑇
6	1, 2, 5	3eqtr3g 2818	. 2 ⊢ ((𝐴 ∖ {𝑥}) = 𝑆 → (𝐴 ∪ {𝑥}) = 𝑇)
7		snssi 4742	. . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
8		ssequn2 4186	. . . 4 ⊢ ({𝑥} ⊆ 𝐴 ↔ (𝐴 ∪ {𝑥}) = 𝐴)
9	7, 8	sylib 221	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐴 ∪ {𝑥}) = 𝐴)
10	9	eqeq1d 2762	. 2 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∪ {𝑥}) = 𝑇 ↔ 𝐴 = 𝑇))
11	6, 10	imbitrid 247	1 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∖ {𝑥}) = 𝑆 → 𝐴 = 𝑇))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   ⊆ wss 3896   ∖ cdif 4044   ∪ cun 4148  {csn 4584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-sn 4585

This theorem is used by:  en2  9279  en3  9280  en4  9281