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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 4399 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
| 2 | 0elon 6437 | . . . 4 ⊢ ∅ ∈ On | |
| 3 | onsseleq 6424 | . . . 4 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴))) | |
| 4 | 2, 3 | mpan 690 | . . 3 ⊢ (𝐴 ∈ On → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴))) | 
| 5 | 1, 4 | mpbii 233 | . 2 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ∨ ∅ = 𝐴)) | 
| 6 | eqcom 2743 | . . . 4 ⊢ (∅ = 𝐴 ↔ 𝐴 = ∅) | |
| 7 | 6 | orbi2i 912 | . . 3 ⊢ ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (∅ ∈ 𝐴 ∨ 𝐴 = ∅)) | 
| 8 | orcom 870 | . . 3 ⊢ ((∅ ∈ 𝐴 ∨ 𝐴 = ∅) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) | |
| 9 | 7, 8 | bitri 275 | . 2 ⊢ ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) | 
| 10 | 5, 9 | sylib 218 | 1 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∨ wo 847 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 ∅c0 4332 Oncon0 6383 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 | 
| This theorem is referenced by: snsn0non 6508 onxpdisj 6509 omabs 8690 cnfcom3lem 9744 0elold 27948 onexlimgt 43260 onexoegt 43261 oe0rif 43303 oege1 43324 onmcl 43349 omabs2 43350 omcl2 43351 | 
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