Theorem sseliALT 5256

Description: Alternate proof of sseli 3931 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3932. (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 262	. 2 ⊢ (𝐴 = if(𝐶 ∈ 𝐴, 𝐴, {∅}) → (𝐶 ∈ 𝐵 ↔ 𝐶 ∈ 𝐵))
2		eleq2 2826	. 2 ⊢ (𝐵 = if(𝐶 ∈ 𝐴, 𝐵, {∅}) → (𝐶 ∈ 𝐵 ↔ 𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐵, {∅})))
3		eleq1 2825	. 2 ⊢ (𝐶 = if(𝐶 ∈ 𝐴, 𝐶, ∅) → (𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐵, {∅}) ↔ if(𝐶 ∈ 𝐴, 𝐶, ∅) ∈ if(𝐶 ∈ 𝐴, 𝐵, {∅})))
4		sseq1 3961	. . . 4 ⊢ (𝐴 = if(𝐶 ∈ 𝐴, 𝐴, {∅}) → (𝐴 ⊆ 𝐵 ↔ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ 𝐵))
5		sseq2 3962	. . . 4 ⊢ (𝐵 = if(𝐶 ∈ 𝐴, 𝐵, {∅}) → (if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ 𝐵 ↔ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ if(𝐶 ∈ 𝐴, 𝐵, {∅})))
6		biidd 262	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ 𝐴, 𝐶, ∅) → (if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ if(𝐶 ∈ 𝐴, 𝐵, {∅}) ↔ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ if(𝐶 ∈ 𝐴, 𝐵, {∅})))
7		sseq1 3961	. . . 4 ⊢ ({∅} = if(𝐶 ∈ 𝐴, 𝐴, {∅}) → ({∅} ⊆ {∅} ↔ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ {∅}))
8		sseq2 3962	. . . 4 ⊢ ({∅} = if(𝐶 ∈ 𝐴, 𝐵, {∅}) → (if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ {∅} ↔ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ if(𝐶 ∈ 𝐴, 𝐵, {∅})))
9		biidd 262	. . . 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 3958	. . . 4 ⊢ {∅} ⊆ {∅}
12	4, 5, 6, 7, 8, 9, 10, 11	keephyp3v 4555	. . 3 ⊢ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ if(𝐶 ∈ 𝐴, 𝐵, {∅})
13		eleq2 2826	. . . 4 ⊢ (𝐴 = if(𝐶 ∈ 𝐴, 𝐴, {∅}) → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})))
14		biidd 262	. . . 4 ⊢ (𝐵 = if(𝐶 ∈ 𝐴, 𝐵, {∅}) → (𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ↔ 𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})))
15		eleq1 2825	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ 𝐴, 𝐶, ∅) → (𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ↔ if(𝐶 ∈ 𝐴, 𝐶, ∅) ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})))
16		eleq2 2826	. . . 4 ⊢ ({∅} = if(𝐶 ∈ 𝐴, 𝐴, {∅}) → (∅ ∈ {∅} ↔ ∅ ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})))
17		biidd 262	. . . 4 ⊢ ({∅} = if(𝐶 ∈ 𝐴, 𝐵, {∅}) → (∅ ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ↔ ∅ ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})))
18		eleq1 2825	. . . 4 ⊢ (∅ = if(𝐶 ∈ 𝐴, 𝐶, ∅) → (∅ ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ↔ if(𝐶 ∈ 𝐴, 𝐶, ∅) ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})))
19		0ex 5254	. . . . 5 ⊢ ∅ ∈ V
20	19	snid 4621	. . . 4 ⊢ ∅ ∈ {∅}
21	13, 14, 15, 16, 17, 18, 20	elimhyp3v 4549	. . 3 ⊢ if(𝐶 ∈ 𝐴, 𝐶, ∅) ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})
22	12, 21	sselii 3932	. 2 ⊢ if(𝐶 ∈ 𝐴, 𝐶, ∅) ∈ if(𝐶 ∈ 𝐴, 𝐵, {∅})
23	1, 2, 3, 22	dedth3v 4545	1 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ⊆ wss 3903  ∅c0 4287  ifcif 4481  {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  ax-nul 5253

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-dif 3906  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583

This theorem is referenced by: (None)