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Theorem sseliALT 4751
Description: Alternate proof of sseli 3579 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3580. (Contributed by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
sseliALT.1 𝐴𝐵
Assertion
Ref Expression
sseliALT (𝐶𝐴𝐶𝐵)

Proof of Theorem sseliALT
StepHypRef Expression
1 biidd 252 . 2 (𝐴 = if(𝐶𝐴, 𝐴, {∅}) → (𝐶𝐵𝐶𝐵))
2 eleq2 2687 . 2 (𝐵 = if(𝐶𝐴, 𝐵, {∅}) → (𝐶𝐵𝐶 ∈ if(𝐶𝐴, 𝐵, {∅})))
3 eleq1 2686 . 2 (𝐶 = if(𝐶𝐴, 𝐶, ∅) → (𝐶 ∈ if(𝐶𝐴, 𝐵, {∅}) ↔ if(𝐶𝐴, 𝐶, ∅) ∈ if(𝐶𝐴, 𝐵, {∅})))
4 sseq1 3605 . . . 4 (𝐴 = if(𝐶𝐴, 𝐴, {∅}) → (𝐴𝐵 ↔ if(𝐶𝐴, 𝐴, {∅}) ⊆ 𝐵))
5 sseq2 3606 . . . 4 (𝐵 = if(𝐶𝐴, 𝐵, {∅}) → (if(𝐶𝐴, 𝐴, {∅}) ⊆ 𝐵 ↔ if(𝐶𝐴, 𝐴, {∅}) ⊆ if(𝐶𝐴, 𝐵, {∅})))
6 biidd 252 . . . 4 (𝐶 = if(𝐶𝐴, 𝐶, ∅) → (if(𝐶𝐴, 𝐴, {∅}) ⊆ if(𝐶𝐴, 𝐵, {∅}) ↔ if(𝐶𝐴, 𝐴, {∅}) ⊆ if(𝐶𝐴, 𝐵, {∅})))
7 sseq1 3605 . . . 4 ({∅} = if(𝐶𝐴, 𝐴, {∅}) → ({∅} ⊆ {∅} ↔ if(𝐶𝐴, 𝐴, {∅}) ⊆ {∅}))
8 sseq2 3606 . . . 4 ({∅} = if(𝐶𝐴, 𝐵, {∅}) → (if(𝐶𝐴, 𝐴, {∅}) ⊆ {∅} ↔ if(𝐶𝐴, 𝐴, {∅}) ⊆ if(𝐶𝐴, 𝐵, {∅})))
9 biidd 252 . . . 4 (∅ = if(𝐶𝐴, 𝐶, ∅) → (if(𝐶𝐴, 𝐴, {∅}) ⊆ if(𝐶𝐴, 𝐵, {∅}) ↔ if(𝐶𝐴, 𝐴, {∅}) ⊆ if(𝐶𝐴, 𝐵, {∅})))
10 sseliALT.1 . . . 4 𝐴𝐵
11 ssid 3603 . . . 4 {∅} ⊆ {∅}
124, 5, 6, 7, 8, 9, 10, 11keephyp3v 4126 . . 3 if(𝐶𝐴, 𝐴, {∅}) ⊆ if(𝐶𝐴, 𝐵, {∅})
13 eleq2 2687 . . . 4 (𝐴 = if(𝐶𝐴, 𝐴, {∅}) → (𝐶𝐴𝐶 ∈ if(𝐶𝐴, 𝐴, {∅})))
14 biidd 252 . . . 4 (𝐵 = if(𝐶𝐴, 𝐵, {∅}) → (𝐶 ∈ if(𝐶𝐴, 𝐴, {∅}) ↔ 𝐶 ∈ if(𝐶𝐴, 𝐴, {∅})))
15 eleq1 2686 . . . 4 (𝐶 = if(𝐶𝐴, 𝐶, ∅) → (𝐶 ∈ if(𝐶𝐴, 𝐴, {∅}) ↔ if(𝐶𝐴, 𝐶, ∅) ∈ if(𝐶𝐴, 𝐴, {∅})))
16 eleq2 2687 . . . 4 ({∅} = if(𝐶𝐴, 𝐴, {∅}) → (∅ ∈ {∅} ↔ ∅ ∈ if(𝐶𝐴, 𝐴, {∅})))
17 biidd 252 . . . 4 ({∅} = if(𝐶𝐴, 𝐵, {∅}) → (∅ ∈ if(𝐶𝐴, 𝐴, {∅}) ↔ ∅ ∈ if(𝐶𝐴, 𝐴, {∅})))
18 eleq1 2686 . . . 4 (∅ = if(𝐶𝐴, 𝐶, ∅) → (∅ ∈ if(𝐶𝐴, 𝐴, {∅}) ↔ if(𝐶𝐴, 𝐶, ∅) ∈ if(𝐶𝐴, 𝐴, {∅})))
19 0ex 4750 . . . . 5 ∅ ∈ V
2019snid 4179 . . . 4 ∅ ∈ {∅}
2113, 14, 15, 16, 17, 18, 20elimhyp3v 4120 . . 3 if(𝐶𝐴, 𝐶, ∅) ∈ if(𝐶𝐴, 𝐴, {∅})
2212, 21sselii 3580 . 2 if(𝐶𝐴, 𝐶, ∅) ∈ if(𝐶𝐴, 𝐵, {∅})
231, 2, 3, 22dedth3v 4116 1 (𝐶𝐴𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wss 3555  c0 3891  ifcif 4058  {csn 4148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-dif 3558  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149
This theorem is referenced by: (None)
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