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Mirrors > Home > MPE Home > Th. List > Mathboxes > naddword1 | Structured version Visualization version GIF version |
Description: Weak-ordering principle for natural addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
Ref | Expression |
---|---|
naddword1 | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +no 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | naddid1 34232 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 +no ∅) = 𝐴) | |
2 | 1 | adantr 481 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no ∅) = 𝐴) |
3 | 0ss 4354 | . . 3 ⊢ ∅ ⊆ 𝐵 | |
4 | 0elon 6369 | . . . . 5 ⊢ ∅ ∈ On | |
5 | naddss2 34238 | . . . . 5 ⊢ ((∅ ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐵 ↔ (𝐴 +no ∅) ⊆ (𝐴 +no 𝐵))) | |
6 | 4, 5 | mp3an1 1448 | . . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐵 ↔ (𝐴 +no ∅) ⊆ (𝐴 +no 𝐵))) |
7 | 6 | ancoms 459 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ⊆ 𝐵 ↔ (𝐴 +no ∅) ⊆ (𝐴 +no 𝐵))) |
8 | 3, 7 | mpbii 232 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no ∅) ⊆ (𝐴 +no 𝐵)) |
9 | 2, 8 | eqsstrrd 3981 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +no 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3908 ∅c0 4280 Oncon0 6315 (class class class)co 7351 +no cnadd 34214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-1st 7913 df-2nd 7914 df-frecs 8204 df-nadd 34215 |
This theorem is referenced by: addsproplem2 34278 |
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