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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 4406 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
2 | 0elon 6440 | . . . 4 ⊢ ∅ ∈ On | |
3 | onsseleq 6427 | . . . 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 2742 | . . . 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 1537 ∈ wcel 2106 ⊆ wss 3963 ∅c0 4339 Oncon0 6386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 |
This theorem is referenced by: snsn0non 6511 onxpdisj 6512 omabs 8688 cnfcom3lem 9741 0elold 27962 onexlimgt 43232 onexoegt 43233 oe0rif 43275 oege1 43296 onmcl 43321 omabs2 43322 omcl2 43323 |
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