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Theorem domnrrg 20205

Description: In a domain, any nonzero element is a nonzero-divisor. (Contributed by Mario Carneiro, 28-Mar-2015.)

**Hypotheses**
Ref	Expression
isdomn2.b	⊢ 𝐵 = (Base‘𝑅)
isdomn2.t	⊢ 𝐸 = (RLReg‘𝑅)
isdomn2.z	⊢ 0 = (0_g‘𝑅)

**Assertion**
Ref	Expression
domnrrg	⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝐸)

**Proof of Theorem domnrrg**
Step	Hyp	Ref	Expression
1		isdomn2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
2		isdomn2.t	. . . . 5 ⊢ 𝐸 = (RLReg‘𝑅)
3		isdomn2.z	. . . . 5 ⊢ 0 = (0_g‘𝑅)
4	1, 2, 3	isdomn2 20204	. . . 4 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))
5	4	simprbi 500	. . 3 ⊢ (𝑅 ∈ Domn → (𝐵 ∖ { 0 }) ⊆ 𝐸)
6	5	3ad2ant1 1134	. 2 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐵 ∖ { 0 }) ⊆ 𝐸)
7		simp2 1138	. . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝐵)
8		simp3 1139	. . 3 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ≠ 0 )
9		eldifsn 4685	. . 3 ⊢ (𝑋 ∈ (𝐵 ∖ { 0 }) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ))
10	7, 8, 9	sylanbrc 586	. 2 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ (𝐵 ∖ { 0 }))
11	6, 10	sseldd 3888	1 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 ∖ cdif 3850 ⊆ wss 3853 {csn 4526 ‘cfv 6350 Basecbs 16599 0_gc0g 16829 NzRingcnzr 20162 RLRegcrlreg 20184 Domncdomn 20185

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-iota 6308 df-fun 6352 df-fv 6358 df-ov 7186 df-rlreg 20188 df-domn 20189

This theorem is referenced by: deg1ldgdomn 24860

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