nnaword1 - Metamath Proof Explorer

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Theorem nnaword1 7863

Description: Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)

**Assertion**
Ref	Expression
nnaword1	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐴 +_𝑜 𝐵))

**Proof of Theorem nnaword1**
Step	Hyp	Ref	Expression
1		nna0 7838	. . 3 ⊢ (𝐴 ∈ ω → (𝐴 +_𝑜 ∅) = 𝐴)
2	1	adantr 466	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +_𝑜 ∅) = 𝐴)
3		0ss 4116	. . 3 ⊢ ∅ ⊆ 𝐵
4		peano1 7232	. . . 4 ⊢ ∅ ∈ ω
5		nnaword 7861	. . . . 5 ⊢ ((∅ ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ⊆ 𝐵 ↔ (𝐴 +_𝑜 ∅) ⊆ (𝐴 +_𝑜 𝐵)))
6	5	3com13 1118	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ ∅ ∈ ω) → (∅ ⊆ 𝐵 ↔ (𝐴 +_𝑜 ∅) ⊆ (𝐴 +_𝑜 𝐵)))
7	4, 6	mp3an3 1561	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∅ ⊆ 𝐵 ↔ (𝐴 +_𝑜 ∅) ⊆ (𝐴 +_𝑜 𝐵)))
8	3, 7	mpbii 223	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +_𝑜 ∅) ⊆ (𝐴 +_𝑜 𝐵))
9	2, 8	eqsstr3d 3789	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐴 +_𝑜 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ⊆ wss 3723 ∅c0 4063 (class class class)co 6793 ωcom 7212 +_𝑜 coa 7710

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-oadd 7717

This theorem is referenced by: nnaword2 7864 nnmordi 7865 nnawordex 7871 omopthlem2 7890 unfilem1 8380 unfi 8383 ackbij1lem12 9255