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Theorem pwtpss 3781
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.
StepHypRef Expression
1 sstpr 3732 . . 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 3259 . . . 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 3259 . . . . . 6 (𝑥 ∈ ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ↔ (𝑥 ∈ {∅, {𝐴}} ∨ 𝑥 ∈ {{𝐵}, {𝐴, 𝐵}}))
4 vex 2725 . . . . . . . 8 𝑥 ∈ V
54elpr 3592 . . . . . . 7 (𝑥 ∈ {∅, {𝐴}} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
64elpr 3592 . . . . . . 7 (𝑥 ∈ {{𝐵}, {𝐴, 𝐵}} ↔ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵}))
75, 6orbi12i 754 . . . . . 6 ((𝑥 ∈ {∅, {𝐴}} ∨ 𝑥 ∈ {{𝐵}, {𝐴, 𝐵}}) ↔ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})))
83, 7bitri 183 . . . . 5 (𝑥 ∈ ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ↔ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})))
9 elun 3259 . . . . . 6 (𝑥 ∈ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}}) ↔ (𝑥 ∈ {{𝐶}, {𝐴, 𝐶}} ∨ 𝑥 ∈ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}}))
104elpr 3592 . . . . . . 7 (𝑥 ∈ {{𝐶}, {𝐴, 𝐶}} ↔ (𝑥 = {𝐶} ∨ 𝑥 = {𝐴, 𝐶}))
114elpr 3592 . . . . . . 7 (𝑥 ∈ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}} ↔ (𝑥 = {𝐵, 𝐶} ∨ 𝑥 = {𝐴, 𝐵, 𝐶}))
1210, 11orbi12i 754 . . . . . 6 ((𝑥 ∈ {{𝐶}, {𝐴, 𝐶}} ∨ 𝑥 ∈ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}}) ↔ ((𝑥 = {𝐶} ∨ 𝑥 = {𝐴, 𝐶}) ∨ (𝑥 = {𝐵, 𝐶} ∨ 𝑥 = {𝐴, 𝐵, 𝐶})))
139, 12bitri 183 . . . . 5 (𝑥 ∈ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}}) ↔ ((𝑥 = {𝐶} ∨ 𝑥 = {𝐴, 𝐶}) ∨ (𝑥 = {𝐵, 𝐶} ∨ 𝑥 = {𝐴, 𝐵, 𝐶})))
148, 13orbi12i 754 . . . 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}))))
152, 14bitri 183 . . 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}))))
164elpw 3560 . . 3 (𝑥 ∈ 𝒫 {𝐴, 𝐵, 𝐶} ↔ 𝑥 ⊆ {𝐴, 𝐵, 𝐶})
171, 15, 163imtr4i 200 . 2 (𝑥 ∈ (({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∪ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}})) → 𝑥 ∈ 𝒫 {𝐴, 𝐵, 𝐶})
1817ssriv 3142 1 (({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∪ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}})) ⊆ 𝒫 {𝐴, 𝐵, 𝐶}
Colors of variables: wff set class
Syntax hints:  wo 698   = wceq 1342  wcel 2135  cun 3110  wss 3112  c0 3405  𝒫 cpw 3554  {csn 3571  {cpr 3572  {ctp 3573
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-3or 968  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2724  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-nul 3406  df-pw 3556  df-sn 3577  df-pr 3578  df-tp 3579
This theorem is referenced by: (None)
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