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Mirrors > Home > MPE Home > Th. List > on0eqel | Structured version Visualization version GIF version |
Description: An ordinal number either equals zero or contains zero. (Contributed by NM, 1-Jun-2004.) |
Ref | Expression |
---|---|
on0eqel | ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4354 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
2 | 0elon 6369 | . . . 4 ⊢ ∅ ∈ On | |
3 | onsseleq 6356 | . . . 4 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴))) | |
4 | 2, 3 | mpan 688 | . . 3 ⊢ (𝐴 ∈ On → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴))) |
5 | 1, 4 | mpbii 232 | . 2 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ∨ ∅ = 𝐴)) |
6 | eqcom 2743 | . . . 4 ⊢ (∅ = 𝐴 ↔ 𝐴 = ∅) | |
7 | 6 | orbi2i 911 | . . 3 ⊢ ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (∅ ∈ 𝐴 ∨ 𝐴 = ∅)) |
8 | orcom 868 | . . 3 ⊢ ((∅ ∈ 𝐴 ∨ 𝐴 = ∅) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) | |
9 | 7, 8 | bitri 274 | . 2 ⊢ ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) |
10 | 5, 9 | sylib 217 | 1 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ⊆ wss 3908 ∅c0 4280 Oncon0 6315 |
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-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
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-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 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-br 5104 df-opab 5166 df-tr 5221 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-ord 6318 df-on 6319 |
This theorem is referenced by: snsn0non 6439 onxpdisj 6440 omabs 8593 cnfcom3lem 9635 onexlimgt 41515 onexoegt 41516 oe0rif 41558 oege1 41578 omabs2 41603 omcl2 41604 |
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