Theorem sstpr 3744

Description: The subsets of a triple. (Contributed by Jim Kingdon, 11-Aug-2018.)

Assertion

Ref	Expression
sstpr	⊢ ((((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) ∨ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}))) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})

Proof of Theorem sstpr

Step	Hyp	Ref	Expression
1		ssprr 3743	. . 3 ⊢ (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶})
2		prsstp12 3733	. . 3 ⊢ {𝐵, 𝐶} ⊆ {𝐵, 𝐶, 𝐷}
3	1, 2	sstrdi 3159	. 2 ⊢ (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
4		snsstp3 3732	. . . . 5 ⊢ {𝐷} ⊆ {𝐵, 𝐶, 𝐷}
5		sseq1 3170	. . . . 5 ⊢ (𝐴 = {𝐷} → (𝐴 ⊆ {𝐵, 𝐶, 𝐷} ↔ {𝐷} ⊆ {𝐵, 𝐶, 𝐷}))
6	4, 5	mpbiri 167	. . . 4 ⊢ (𝐴 = {𝐷} → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
7		prsstp13 3734	. . . . 5 ⊢ {𝐵, 𝐷} ⊆ {𝐵, 𝐶, 𝐷}
8		sseq1 3170	. . . . 5 ⊢ (𝐴 = {𝐵, 𝐷} → (𝐴 ⊆ {𝐵, 𝐶, 𝐷} ↔ {𝐵, 𝐷} ⊆ {𝐵, 𝐶, 𝐷}))
9	7, 8	mpbiri 167	. . . 4 ⊢ (𝐴 = {𝐵, 𝐷} → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
10	6, 9	jaoi 711	. . 3 ⊢ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
11		prsstp23 3735	. . . . 5 ⊢ {𝐶, 𝐷} ⊆ {𝐵, 𝐶, 𝐷}
12		sseq1 3170	. . . . 5 ⊢ (𝐴 = {𝐶, 𝐷} → (𝐴 ⊆ {𝐵, 𝐶, 𝐷} ↔ {𝐶, 𝐷} ⊆ {𝐵, 𝐶, 𝐷}))
13	11, 12	mpbiri 167	. . . 4 ⊢ (𝐴 = {𝐶, 𝐷} → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
14		eqimss 3201	. . . 4 ⊢ (𝐴 = {𝐵, 𝐶, 𝐷} → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
15	13, 14	jaoi 711	. . 3 ⊢ ((𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
16	10, 15	jaoi 711	. 2 ⊢ (((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷})) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
17	3, 16	jaoi 711	1 ⊢ ((((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) ∨ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}))) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})

Syntax hints:   → wi 4   ∨ wo 703   = wceq 1348   ⊆ wss 3121  ∅c0 3414  {csn 3583  {cpr 3584  {ctp 3585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3or 974  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-pr 3590  df-tp 3591

This theorem is referenced by:  pwtpss  3793