Theorem sstpr 3684

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 3683	. . 3 ⊢ (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶})
2		prsstp12 3673	. . 3 ⊢ {𝐵, 𝐶} ⊆ {𝐵, 𝐶, 𝐷}
3	1, 2	sstrdi 3109	. 2 ⊢ (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
4		snsstp3 3672	. . . . 5 ⊢ {𝐷} ⊆ {𝐵, 𝐶, 𝐷}
5		sseq1 3120	. . . . 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 3674	. . . . 5 ⊢ {𝐵, 𝐷} ⊆ {𝐵, 𝐶, 𝐷}
8		sseq1 3120	. . . . 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 705	. . 3 ⊢ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
11		prsstp23 3675	. . . . 5 ⊢ {𝐶, 𝐷} ⊆ {𝐵, 𝐶, 𝐷}
12		sseq1 3120	. . . . 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 3151	. . . 4 ⊢ (𝐴 = {𝐵, 𝐶, 𝐷} → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
15	13, 14	jaoi 705	. . 3 ⊢ ((𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
16	10, 15	jaoi 705	. 2 ⊢ (((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷})) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
17	3, 16	jaoi 705	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 697   = wceq 1331   ⊆ wss 3071  ∅c0 3363  {csn 3527  {cpr 3528  {ctp 3529

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3or 963  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-pr 3534  df-tp 3535

This theorem is referenced by:  pwtpss  3733