Theorem sseliALT 5265

Description: Alternate proof of sseli 3908 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3909. (Contributed by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sseliALT.1	⊢ 𝐴 ⊆ 𝐵

Assertion

Ref	Expression
sseliALT	⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)

Proof of Theorem sseliALT

Step	Hyp	Ref	Expression
1		biidd 265	. 2 ⊢ (𝐴 = if(𝐶 ∈ 𝐴, 𝐴, {∅}) → (𝐶 ∈ 𝐵 ↔ 𝐶 ∈ 𝐵))
2		eleq2 2852	. 2 ⊢ (𝐵 = if(𝐶 ∈ 𝐴, 𝐵, {∅}) → (𝐶 ∈ 𝐵 ↔ 𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐵, {∅})))
3		eleq1 2851	. 2 ⊢ (𝐶 = if(𝐶 ∈ 𝐴, 𝐶, ∅) → (𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐵, {∅}) ↔ if(𝐶 ∈ 𝐴, 𝐶, ∅) ∈ if(𝐶 ∈ 𝐴, 𝐵, {∅})))
4		sseq1 3937	. . . 4 ⊢ (𝐴 = if(𝐶 ∈ 𝐴, 𝐴, {∅}) → (𝐴 ⊆ 𝐵 ↔ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ 𝐵))
5		sseq2 3938	. . . 4 ⊢ (𝐵 = if(𝐶 ∈ 𝐴, 𝐵, {∅}) → (if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ 𝐵 ↔ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ if(𝐶 ∈ 𝐴, 𝐵, {∅})))
6		biidd 265	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ 𝐴, 𝐶, ∅) → (if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ if(𝐶 ∈ 𝐴, 𝐵, {∅}) ↔ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ if(𝐶 ∈ 𝐴, 𝐵, {∅})))
7		sseq1 3937	. . . 4 ⊢ ({∅} = if(𝐶 ∈ 𝐴, 𝐴, {∅}) → ({∅} ⊆ {∅} ↔ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ {∅}))
8		sseq2 3938	. . . 4 ⊢ ({∅} = if(𝐶 ∈ 𝐴, 𝐵, {∅}) → (if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ {∅} ↔ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ if(𝐶 ∈ 𝐴, 𝐵, {∅})))
9		biidd 265	. . . 4 ⊢ (∅ = if(𝐶 ∈ 𝐴, 𝐶, ∅) → (if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ if(𝐶 ∈ 𝐴, 𝐵, {∅}) ↔ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ if(𝐶 ∈ 𝐴, 𝐵, {∅})))
10		sseliALT.1	. . . 4 ⊢ 𝐴 ⊆ 𝐵
11		ssid 3934	. . . 4 ⊢ {∅} ⊆ {∅}
12	4, 5, 6, 7, 8, 9, 10, 11	keephyp3v 4558	. . 3 ⊢ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ if(𝐶 ∈ 𝐴, 𝐵, {∅})
13		eleq2 2852	. . . 4 ⊢ (𝐴 = if(𝐶 ∈ 𝐴, 𝐴, {∅}) → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})))
14		biidd 265	. . . 4 ⊢ (𝐵 = if(𝐶 ∈ 𝐴, 𝐵, {∅}) → (𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ↔ 𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})))
15		eleq1 2851	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ 𝐴, 𝐶, ∅) → (𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ↔ if(𝐶 ∈ 𝐴, 𝐶, ∅) ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})))
16		eleq2 2852	. . . 4 ⊢ ({∅} = if(𝐶 ∈ 𝐴, 𝐴, {∅}) → (∅ ∈ {∅} ↔ ∅ ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})))
17		biidd 265	. . . 4 ⊢ ({∅} = if(𝐶 ∈ 𝐴, 𝐵, {∅}) → (∅ ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ↔ ∅ ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})))
18		eleq1 2851	. . . 4 ⊢ (∅ = if(𝐶 ∈ 𝐴, 𝐶, ∅) → (∅ ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ↔ if(𝐶 ∈ 𝐴, 𝐶, ∅) ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})))
19		0ex 5263	. . . . 5 ⊢ ∅ ∈ V
20	19	snid 4622	. . . 4 ⊢ ∅ ∈ {∅}
21	13, 14, 15, 16, 17, 18, 20	elimhyp3v 4552	. . 3 ⊢ if(𝐶 ∈ 𝐴, 𝐶, ∅) ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})
22	12, 21	sselii 3909	. 2 ⊢ if(𝐶 ∈ 𝐴, 𝐶, ∅) ∈ if(𝐶 ∈ 𝐴, 𝐵, {∅})
23	1, 2, 3, 22	dedth3v 4548	1 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2146   ⊆ wss 3899  ∅c0 4280  ifcif 4484  {csn 4587

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-nul 5262

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3455  df-ss 3900  df-dif 4049  df-nul 4281  df-if 4485  df-sn 4588

This theorem is used by: (None)