Theorem sseliALT 5309

Description: Alternate proof of sseli 3979 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3980. (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 2830	. 2 ⊢ (𝐵 = if(𝐶 ∈ 𝐴, 𝐵, {∅}) → (𝐶 ∈ 𝐵 ↔ 𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐵, {∅})))
3		eleq1 2829	. 2 ⊢ (𝐶 = if(𝐶 ∈ 𝐴, 𝐶, ∅) → (𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐵, {∅}) ↔ if(𝐶 ∈ 𝐴, 𝐶, ∅) ∈ if(𝐶 ∈ 𝐴, 𝐵, {∅})))
4		sseq1 4009	. . . 4 ⊢ (𝐴 = if(𝐶 ∈ 𝐴, 𝐴, {∅}) → (𝐴 ⊆ 𝐵 ↔ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ 𝐵))
5		sseq2 4010	. . . 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 4009	. . . 4 ⊢ ({∅} = if(𝐶 ∈ 𝐴, 𝐴, {∅}) → ({∅} ⊆ {∅} ↔ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ {∅}))
8		sseq2 4010	. . . 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 4006	. . . 4 ⊢ {∅} ⊆ {∅}
12	4, 5, 6, 7, 8, 9, 10, 11	keephyp3v 4599	. . 3 ⊢ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ if(𝐶 ∈ 𝐴, 𝐵, {∅})
13		eleq2 2830	. . . 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 2829	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ 𝐴, 𝐶, ∅) → (𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ↔ if(𝐶 ∈ 𝐴, 𝐶, ∅) ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})))
16		eleq2 2830	. . . 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 2829	. . . 4 ⊢ (∅ = if(𝐶 ∈ 𝐴, 𝐶, ∅) → (∅ ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ↔ if(𝐶 ∈ 𝐴, 𝐶, ∅) ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})))
19		0ex 5307	. . . . 5 ⊢ ∅ ∈ V
20	19	snid 4662	. . . 4 ⊢ ∅ ∈ {∅}
21	13, 14, 15, 16, 17, 18, 20	elimhyp3v 4593	. . 3 ⊢ if(𝐶 ∈ 𝐴, 𝐶, ∅) ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})
22	12, 21	sselii 3980	. 2 ⊢ if(𝐶 ∈ 𝐴, 𝐶, ∅) ∈ if(𝐶 ∈ 𝐴, 𝐵, {∅})
23	1, 2, 3, 22	dedth3v 4589	1 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ⊆ wss 3951  ∅c0 4333  ifcif 4525  {csn 4626

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627

This theorem is referenced by: (None)