dmr0ss - Mathbox for BTernaryTau

	Mathbox for BTernaryTau		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dmr0ss		Structured version Visualization version GIF version

Theorem dmr0ss 35934

Description: The domain of 𝑅₀ is a subset of On × On. (Contributed by BTernaryTau, 22-Sep-2026.)

**Assertion**
Ref	Expression
dmr0ss	⊢ dom 𝑅₀ ⊆ (On × On)

Proof of Theorem dmr0ss Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-r0 35922	. . . 4 ⊢ 𝑅₀ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ ((1^st ‘𝑦) ∪ (2^nd ‘𝑦)) ∨ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) = ((1^st ‘𝑦) ∪ (2^nd ‘𝑦)) ∧ 𝑥LexOrd𝑦)))}
2		simpll 779	. . . . . 6 ⊢ (((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ ((1^st ‘𝑦) ∪ (2^nd ‘𝑦)) ∨ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) = ((1^st ‘𝑦) ∪ (2^nd ‘𝑦)) ∧ 𝑥LexOrd𝑦))) → 𝑥 ∈ (On × On))
3	2	pm4.71ri 570	. . . . 5 ⊢ (((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ ((1^st ‘𝑦) ∪ (2^nd ‘𝑦)) ∨ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) = ((1^st ‘𝑦) ∪ (2^nd ‘𝑦)) ∧ 𝑥LexOrd𝑦))) ↔ (𝑥 ∈ (On × On) ∧ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ ((1^st ‘𝑦) ∪ (2^nd ‘𝑦)) ∨ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) = ((1^st ‘𝑦) ∪ (2^nd ‘𝑦)) ∧ 𝑥LexOrd𝑦)))))
4	3	opabbii 5171	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ ((1^st ‘𝑦) ∪ (2^nd ‘𝑦)) ∨ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) = ((1^st ‘𝑦) ∪ (2^nd ‘𝑦)) ∧ 𝑥LexOrd𝑦)))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ ((1^st ‘𝑦) ∪ (2^nd ‘𝑦)) ∨ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) = ((1^st ‘𝑦) ∪ (2^nd ‘𝑦)) ∧ 𝑥LexOrd𝑦))))}
5	1, 4	eqtri 2783	. . 3 ⊢ 𝑅₀ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ ((1^st ‘𝑦) ∪ (2^nd ‘𝑦)) ∨ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) = ((1^st ‘𝑦) ∪ (2^nd ‘𝑦)) ∧ 𝑥LexOrd𝑦))))}
6	5	dmeqi 5885	. 2 ⊢ dom 𝑅₀ = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ ((1^st ‘𝑦) ∪ (2^nd ‘𝑦)) ∨ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) = ((1^st ‘𝑦) ∪ (2^nd ‘𝑦)) ∧ 𝑥LexOrd𝑦))))}
7		dmopabss 5899	. 2 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) ∈ ((1^st ‘𝑦) ∪ (2^nd ‘𝑦)) ∨ (((1^st ‘𝑥) ∪ (2^nd ‘𝑥)) = ((1^st ‘𝑦) ∪ (2^nd ‘𝑦)) ∧ 𝑥LexOrd𝑦))))} ⊆ (On × On)
8	6, 7	eqsstri 3954	1 ⊢ dom 𝑅₀ ⊆ (On × On)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2144 ⊆ wss 3895 ∪ cun 4149 class class class wbr 5102 {copab 5166 × cxp 5648 dom cdm 5650 Oncon0 6362 ‘cfv 6538 1^st c1st 7999 2^nd c2nd 8000 LexOrdclexo 35912 𝑅₀cr0 35913

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-pr 5390

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-rab 3413 df-v 3452 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-sn 4584 df-pr 4586 df-op 4590 df-br 5103 df-opab 5167 df-dm 5660 df-r0 35922

This theorem is used by: j0ltalephlem 35939

W3C validator