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Mirrors > Home > MPE Home > Th. List > oa00 | Structured version Visualization version GIF version |
Description: An ordinal sum is zero iff both of its arguments are zero. (Contributed by NM, 6-Dec-2004.) |
Ref | Expression |
---|---|
oa00 | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 = ∅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | on0eln0 5697 | . . . . . . 7 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | |
2 | 1 | adantr 480 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
3 | oaword1 7519 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +𝑜 𝐵)) | |
4 | 3 | sseld 3567 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐴 → ∅ ∈ (𝐴 +𝑜 𝐵))) |
5 | 2, 4 | sylbird 249 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≠ ∅ → ∅ ∈ (𝐴 +𝑜 𝐵))) |
6 | ne0i 3880 | . . . . 5 ⊢ (∅ ∈ (𝐴 +𝑜 𝐵) → (𝐴 +𝑜 𝐵) ≠ ∅) | |
7 | 5, 6 | syl6 34 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≠ ∅ → (𝐴 +𝑜 𝐵) ≠ ∅)) |
8 | 7 | necon4d 2806 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) = ∅ → 𝐴 = ∅)) |
9 | on0eln0 5697 | . . . . . . 7 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅)) | |
10 | 9 | adantl 481 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅)) |
11 | 0elon 5695 | . . . . . . . 8 ⊢ ∅ ∈ On | |
12 | oaord 7514 | . . . . . . . 8 ⊢ ((∅ ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐵 ↔ (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵))) | |
13 | 11, 12 | mp3an1 1403 | . . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐵 ↔ (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵))) |
14 | 13 | ancoms 468 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 ↔ (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵))) |
15 | 10, 14 | bitr3d 269 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ≠ ∅ ↔ (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵))) |
16 | ne0i 3880 | . . . . 5 ⊢ ((𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝐵) → (𝐴 +𝑜 𝐵) ≠ ∅) | |
17 | 15, 16 | syl6bi 242 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ≠ ∅ → (𝐴 +𝑜 𝐵) ≠ ∅)) |
18 | 17 | necon4d 2806 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) = ∅ → 𝐵 = ∅)) |
19 | 8, 18 | jcad 554 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅))) |
20 | oveq12 6558 | . . 3 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 +𝑜 𝐵) = (∅ +𝑜 ∅)) | |
21 | oa0 7483 | . . . 4 ⊢ (∅ ∈ On → (∅ +𝑜 ∅) = ∅) | |
22 | 11, 21 | ax-mp 5 | . . 3 ⊢ (∅ +𝑜 ∅) = ∅ |
23 | 20, 22 | syl6eq 2660 | . 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 +𝑜 𝐵) = ∅) |
24 | 19, 23 | impbid1 214 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 = ∅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 Oncon0 5640 (class class class)co 6549 +𝑜 coa 7444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-oadd 7451 |
This theorem is referenced by: oalimcl 7527 oeoa 7564 |
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