Theorem elpg 47599

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 {𝐿 ∣ 𝑅}. 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) ∧ (1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg))

Proof of Theorem elpg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpglem1 47596	. . . 4 ⊢ (∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)) → ((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg))
2		elpglem2 47597	. . . 4 ⊢ (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)))
3	1, 2	impbii 208	. . 3 ⊢ (∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)) ↔ ((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg))
4	3	anbi2i 624	. 2 ⊢ ((𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg)))
5		df-pg 47595	. . . 4 ⊢ Pg = setrecs((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦)))
6	5	elsetrecs 47585	. . 3 ⊢ (𝐴 ∈ Pg ↔ ∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)))
7		elpglem3 47598	. . 3 ⊢ (∃𝑥(𝑥 ⊆ Pg ∧ 𝐴 ∈ ((𝑦 ∈ V ↦ (𝒫 𝑦 × 𝒫 𝑦))‘𝑥)) ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))))
8	6, 7	bitri 275	. 2 ⊢ (𝐴 ∈ Pg ↔ (𝐴 ∈ (V × V) ∧ ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))))
9		3anass 1096	. 2 ⊢ ((𝐴 ∈ (V × V) ∧ (1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg)))
10	4, 8, 9	3bitr4i 303	1 ⊢ (𝐴 ∈ Pg ↔ (𝐴 ∈ (V × V) ∧ (1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg))

Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088  ∃wex 1782   ∈ wcel 2107  Vcvv 3475   ⊆ wss 3946  𝒫 cpw 4598   ↦ cmpt 5227   × cxp 5670  ‘cfv 6535  1^st c1st 7960  2^nd c2nd 7961  Pgcpg 47594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712  ax-reg 9574  ax-inf2 9623

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-int 4947  df-iun 4995  df-iin 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7399  df-om 7843  df-1st 7962  df-2nd 7963  df-frecs 8253  df-wrecs 8284  df-recs 8358  df-rdg 8397  df-r1 9746  df-rank 9747  df-setrecs 47569  df-pg 47595

This theorem is referenced by: (None)