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Theorem elpg 48945
Description: Membership in the class of partisan games. In John Horton Conway's On Numbers and Games, this is stated as "If 𝐿 and 𝑅 are any two sets of games, then there is a game {𝐿𝑅}. All games are constructed in this way." The first sentence corresponds to the backward direction of our theorem, and the second to the forward direction. (Contributed by Emmett Weisz, 27-Aug-2021.)
Assertion
Ref Expression
elpg (𝐴 ∈ Pg ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg))

Proof of Theorem elpg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpglem1 48942 . . . 4 (∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)) → ((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg))
2 elpglem2 48943 . . . 4 (((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)))
31, 2impbii 209 . . 3 (∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)) ↔ ((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg))
43anbi2i 623 . 2 ((𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg)))
5 df-pg 48941 . . . 4 Pg = setrecs((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦)))
65elsetrecs 48931 . . 3 (𝐴 ∈ Pg ↔ ∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)))
7 elpglem3 48944 . . 3 (∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))))
86, 7bitri 275 . 2 (𝐴 ∈ Pg ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))))
9 3anass 1094 . 2 ((𝐴 ∈ (V × V) ∧ (1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg)))
104, 8, 93bitr4i 303 1 (𝐴 ∈ Pg ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086  wex 1776  wcel 2106  Vcvv 3478  wss 3963  𝒫 cpw 4605  cmpt 5231   × cxp 5687  cfv 6563  1st c1st 8011  2nd c2nd 8012  Pgcpg 48940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-r1 9802  df-rank 9803  df-setrecs 48915  df-pg 48941
This theorem is referenced by: (None)
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