Theorem sseliALT 4719

Description: Alternate proof of sseli 3564 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3565. (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 251	. 2 ⊢ (𝐴 = if(𝐶 ∈ 𝐴, 𝐴, {∅}) → (𝐶 ∈ 𝐵 ↔ 𝐶 ∈ 𝐵))
2		eleq2 2677	. 2 ⊢ (𝐵 = if(𝐶 ∈ 𝐴, 𝐵, {∅}) → (𝐶 ∈ 𝐵 ↔ 𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐵, {∅})))
3		eleq1 2676	. 2 ⊢ (𝐶 = if(𝐶 ∈ 𝐴, 𝐶, ∅) → (𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐵, {∅}) ↔ if(𝐶 ∈ 𝐴, 𝐶, ∅) ∈ if(𝐶 ∈ 𝐴, 𝐵, {∅})))
4		sseq1 3589	. . . 4 ⊢ (𝐴 = if(𝐶 ∈ 𝐴, 𝐴, {∅}) → (𝐴 ⊆ 𝐵 ↔ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ 𝐵))
5		sseq2 3590	. . . 4 ⊢ (𝐵 = if(𝐶 ∈ 𝐴, 𝐵, {∅}) → (if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ 𝐵 ↔ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ if(𝐶 ∈ 𝐴, 𝐵, {∅})))
6		biidd 251	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ 𝐴, 𝐶, ∅) → (if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ if(𝐶 ∈ 𝐴, 𝐵, {∅}) ↔ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ if(𝐶 ∈ 𝐴, 𝐵, {∅})))
7		sseq1 3589	. . . 4 ⊢ ({∅} = if(𝐶 ∈ 𝐴, 𝐴, {∅}) → ({∅} ⊆ {∅} ↔ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ {∅}))
8		sseq2 3590	. . . 4 ⊢ ({∅} = if(𝐶 ∈ 𝐴, 𝐵, {∅}) → (if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ {∅} ↔ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ if(𝐶 ∈ 𝐴, 𝐵, {∅})))
9		biidd 251	. . . 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 3587	. . . 4 ⊢ {∅} ⊆ {∅}
12	4, 5, 6, 7, 8, 9, 10, 11	keephyp3v 4104	. . 3 ⊢ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ⊆ if(𝐶 ∈ 𝐴, 𝐵, {∅})
13		eleq2 2677	. . . 4 ⊢ (𝐴 = if(𝐶 ∈ 𝐴, 𝐴, {∅}) → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})))
14		biidd 251	. . . 4 ⊢ (𝐵 = if(𝐶 ∈ 𝐴, 𝐵, {∅}) → (𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ↔ 𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})))
15		eleq1 2676	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ 𝐴, 𝐶, ∅) → (𝐶 ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ↔ if(𝐶 ∈ 𝐴, 𝐶, ∅) ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})))
16		eleq2 2677	. . . 4 ⊢ ({∅} = if(𝐶 ∈ 𝐴, 𝐴, {∅}) → (∅ ∈ {∅} ↔ ∅ ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})))
17		biidd 251	. . . 4 ⊢ ({∅} = if(𝐶 ∈ 𝐴, 𝐵, {∅}) → (∅ ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ↔ ∅ ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})))
18		eleq1 2676	. . . 4 ⊢ (∅ = if(𝐶 ∈ 𝐴, 𝐶, ∅) → (∅ ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅}) ↔ if(𝐶 ∈ 𝐴, 𝐶, ∅) ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})))
19		0ex 4718	. . . . 5 ⊢ ∅ ∈ V
20	19	snid 4155	. . . 4 ⊢ ∅ ∈ {∅}
21	13, 14, 15, 16, 17, 18, 20	elimhyp3v 4098	. . 3 ⊢ if(𝐶 ∈ 𝐴, 𝐶, ∅) ∈ if(𝐶 ∈ 𝐴, 𝐴, {∅})
22	12, 21	sselii 3565	. 2 ⊢ if(𝐶 ∈ 𝐴, 𝐶, ∅) ∈ if(𝐶 ∈ 𝐴, 𝐵, {∅})
23	1, 2, 3, 22	dedth3v 4094	1 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ∅c0 3874  ifcif 4036  {csn 4125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126

This theorem is referenced by: (None)