Theorem pwtpss 3808

Description: The power set of an unordered triple. (Contributed by Jim Kingdon, 13-Aug-2018.)

Assertion

Ref	Expression
pwtpss	⊢ (({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∪ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}})) ⊆ 𝒫 {𝐴, 𝐵, 𝐶}

Proof of Theorem pwtpss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sstpr 3759	. . 3 ⊢ ((((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})) ∨ ((𝑥 = {𝐶} ∨ 𝑥 = {𝐴, 𝐶}) ∨ (𝑥 = {𝐵, 𝐶} ∨ 𝑥 = {𝐴, 𝐵, 𝐶}))) → 𝑥 ⊆ {𝐴, 𝐵, 𝐶})
2		elun 3278	. . . 4 ⊢ (𝑥 ∈ (({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∪ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}})) ↔ (𝑥 ∈ ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∨ 𝑥 ∈ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}})))
3		elun 3278	. . . . . 6 ⊢ (𝑥 ∈ ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ↔ (𝑥 ∈ {∅, {𝐴}} ∨ 𝑥 ∈ {{𝐵}, {𝐴, 𝐵}}))
4		vex 2742	. . . . . . . 8 ⊢ 𝑥 ∈ V
5	4	elpr 3615	. . . . . . 7 ⊢ (𝑥 ∈ {∅, {𝐴}} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
6	4	elpr 3615	. . . . . . 7 ⊢ (𝑥 ∈ {{𝐵}, {𝐴, 𝐵}} ↔ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵}))
7	5, 6	orbi12i 764	. . . . . 6 ⊢ ((𝑥 ∈ {∅, {𝐴}} ∨ 𝑥 ∈ {{𝐵}, {𝐴, 𝐵}}) ↔ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})))
8	3, 7	bitri 184	. . . . 5 ⊢ (𝑥 ∈ ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ↔ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})))
9		elun 3278	. . . . . 6 ⊢ (𝑥 ∈ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}}) ↔ (𝑥 ∈ {{𝐶}, {𝐴, 𝐶}} ∨ 𝑥 ∈ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}}))
10	4	elpr 3615	. . . . . . 7 ⊢ (𝑥 ∈ {{𝐶}, {𝐴, 𝐶}} ↔ (𝑥 = {𝐶} ∨ 𝑥 = {𝐴, 𝐶}))
11	4	elpr 3615	. . . . . . 7 ⊢ (𝑥 ∈ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}} ↔ (𝑥 = {𝐵, 𝐶} ∨ 𝑥 = {𝐴, 𝐵, 𝐶}))
12	10, 11	orbi12i 764	. . . . . 6 ⊢ ((𝑥 ∈ {{𝐶}, {𝐴, 𝐶}} ∨ 𝑥 ∈ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}}) ↔ ((𝑥 = {𝐶} ∨ 𝑥 = {𝐴, 𝐶}) ∨ (𝑥 = {𝐵, 𝐶} ∨ 𝑥 = {𝐴, 𝐵, 𝐶})))
13	9, 12	bitri 184	. . . . 5 ⊢ (𝑥 ∈ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}}) ↔ ((𝑥 = {𝐶} ∨ 𝑥 = {𝐴, 𝐶}) ∨ (𝑥 = {𝐵, 𝐶} ∨ 𝑥 = {𝐴, 𝐵, 𝐶})))
14	8, 13	orbi12i 764	. . . 4 ⊢ ((𝑥 ∈ ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∨ 𝑥 ∈ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}})) ↔ (((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})) ∨ ((𝑥 = {𝐶} ∨ 𝑥 = {𝐴, 𝐶}) ∨ (𝑥 = {𝐵, 𝐶} ∨ 𝑥 = {𝐴, 𝐵, 𝐶}))))
15	2, 14	bitri 184	. . 3 ⊢ (𝑥 ∈ (({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∪ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}})) ↔ (((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})) ∨ ((𝑥 = {𝐶} ∨ 𝑥 = {𝐴, 𝐶}) ∨ (𝑥 = {𝐵, 𝐶} ∨ 𝑥 = {𝐴, 𝐵, 𝐶}))))
16	4	elpw 3583	. . 3 ⊢ (𝑥 ∈ 𝒫 {𝐴, 𝐵, 𝐶} ↔ 𝑥 ⊆ {𝐴, 𝐵, 𝐶})
17	1, 15, 16	3imtr4i 201	. 2 ⊢ (𝑥 ∈ (({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∪ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}})) → 𝑥 ∈ 𝒫 {𝐴, 𝐵, 𝐶})
18	17	ssriv 3161	1 ⊢ (({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∪ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}})) ⊆ 𝒫 {𝐴, 𝐵, 𝐶}

Syntax hints:   ∨ wo 708   = wceq 1353   ∈ wcel 2148   ∪ cun 3129   ⊆ wss 3131  ∅c0 3424  𝒫 cpw 3577  {csn 3594  {cpr 3595  {ctp 3596

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3or 979  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-tp 3602

This theorem is referenced by: (None)