Theorem sstpr 3798

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 3797	. . 3 ⊢ (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶})
2		prsstp12 3786	. . 3 ⊢ {𝐵, 𝐶} ⊆ {𝐵, 𝐶, 𝐷}
3	1, 2	sstrdi 3205	. 2 ⊢ (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
4		snsstp3 3785	. . . . 5 ⊢ {𝐷} ⊆ {𝐵, 𝐶, 𝐷}
5		sseq1 3216	. . . . 5 ⊢ (𝐴 = {𝐷} → (𝐴 ⊆ {𝐵, 𝐶, 𝐷} ↔ {𝐷} ⊆ {𝐵, 𝐶, 𝐷}))
6	4, 5	mpbiri 168	. . . 4 ⊢ (𝐴 = {𝐷} → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
7		prsstp13 3787	. . . . 5 ⊢ {𝐵, 𝐷} ⊆ {𝐵, 𝐶, 𝐷}
8		sseq1 3216	. . . . 5 ⊢ (𝐴 = {𝐵, 𝐷} → (𝐴 ⊆ {𝐵, 𝐶, 𝐷} ↔ {𝐵, 𝐷} ⊆ {𝐵, 𝐶, 𝐷}))
9	7, 8	mpbiri 168	. . . 4 ⊢ (𝐴 = {𝐵, 𝐷} → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
10	6, 9	jaoi 718	. . 3 ⊢ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
11		prsstp23 3788	. . . . 5 ⊢ {𝐶, 𝐷} ⊆ {𝐵, 𝐶, 𝐷}
12		sseq1 3216	. . . . 5 ⊢ (𝐴 = {𝐶, 𝐷} → (𝐴 ⊆ {𝐵, 𝐶, 𝐷} ↔ {𝐶, 𝐷} ⊆ {𝐵, 𝐶, 𝐷}))
13	11, 12	mpbiri 168	. . . 4 ⊢ (𝐴 = {𝐶, 𝐷} → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
14		eqimss 3247	. . . 4 ⊢ (𝐴 = {𝐵, 𝐶, 𝐷} → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
15	13, 14	jaoi 718	. . 3 ⊢ ((𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
16	10, 15	jaoi 718	. 2 ⊢ (((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷})) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
17	3, 16	jaoi 718	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 710   = wceq 1373   ⊆ wss 3166  ∅c0 3460  {csn 3633  {cpr 3634  {ctp 3635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3or 982  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-sn 3639  df-pr 3640  df-tp 3641

This theorem is referenced by:  pwtpss  3847