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Theorem elpg 47749
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 47746 . . . 4 (∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)) → ((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg))
2 elpglem2 47747 . . . 4 (((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)))
31, 2impbii 208 . . 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 47745 . . . 4 Pg = setrecs((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦)))
65elsetrecs 47735 . . 3 (𝐴 ∈ Pg ↔ ∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)))
7 elpglem3 47748 . . 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 274 . 2 (𝐴 ∈ Pg ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))))
9 3anass 1095 . 2 ((𝐴 ∈ (V × V) ∧ (1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) ↔ (𝐴 ∈ (V × V) ∧ ((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg)))
104, 8, 93bitr4i 302 1 (𝐴 ∈ Pg ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  w3a 1087  wex 1781  wcel 2106  Vcvv 3474  wss 3948  𝒫 cpw 4602  cmpt 5231   × cxp 5674  cfv 6543  1st c1st 7972  2nd c2nd 7973  Pgcpg 47744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-reg 9586  ax-inf2 9635
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-r1 9758  df-rank 9759  df-setrecs 47719  df-pg 47745
This theorem is referenced by: (None)
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