Theorem goeleq12bg 35361

Description: Two "Godel-set of membership" codes for two variables are equal iff the two corresponding variables are equal. (Contributed by AV, 8-Oct-2023.)

Assertion

Ref	Expression
goeleq12bg	⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼∈𝑔𝐽) = (𝑀∈𝑔𝑁) ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁)))

Proof of Theorem goeleq12bg

Step	Hyp	Ref	Expression
1		goel 35359	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
2		goel 35359	. . 3 ⊢ ((𝑀 ∈ ω ∧ 𝑁 ∈ ω) → (𝑀∈𝑔𝑁) = ⟨∅, ⟨𝑀, 𝑁⟩⟩)
3	1, 2	eqeqan12rd 2745	. 2 ⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼∈𝑔𝐽) = (𝑀∈𝑔𝑁) ↔ ⟨∅, ⟨𝐼, 𝐽⟩⟩ = ⟨∅, ⟨𝑀, 𝑁⟩⟩))
4		0ex 5243	. . . 4 ⊢ ∅ ∈ V
5		opex 5402	. . . 4 ⊢ ⟨𝐼, 𝐽⟩ ∈ V
6	4, 5	opth 5414	. . 3 ⊢ (⟨∅, ⟨𝐼, 𝐽⟩⟩ = ⟨∅, ⟨𝑀, 𝑁⟩⟩ ↔ (∅ = ∅ ∧ ⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩))
7		eqid 2730	. . . . 5 ⊢ ∅ = ∅
8	7	biantrur 530	. . . 4 ⊢ (⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩ ↔ (∅ = ∅ ∧ ⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩))
9		opthg 5415	. . . . 5 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩ ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁)))
10	9	adantl 481	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → (⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩ ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁)))
11	8, 10	bitr3id 285	. . 3 ⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((∅ = ∅ ∧ ⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩) ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁)))
12	6, 11	bitrid 283	. 2 ⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → (⟨∅, ⟨𝐼, 𝐽⟩⟩ = ⟨∅, ⟨𝑀, 𝑁⟩⟩ ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁)))
13	3, 12	bitrd 279	1 ⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼∈𝑔𝐽) = (𝑀∈𝑔𝑁) ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110  ∅c0 4281  ⟨cop 4580  (class class class)co 7341  ωcom 7791  ∈_𝑔cgoe 35345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6433  df-fun 6479  df-fv 6485  df-ov 7344  df-goel 35352

This theorem is referenced by:  satfv0  35370  satfv0fun  35383