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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 4350 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
2 | 0elon 6239 | . . . 4 ⊢ ∅ ∈ On | |
3 | onsseleq 6227 | . . . 4 ⊢ ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴))) | |
4 | 2, 3 | mpan 688 | . . 3 ⊢ (𝐴 ∈ On → (∅ ⊆ 𝐴 ↔ (∅ ∈ 𝐴 ∨ ∅ = 𝐴))) |
5 | 1, 4 | mpbii 235 | . 2 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ∨ ∅ = 𝐴)) |
6 | eqcom 2828 | . . . 4 ⊢ (∅ = 𝐴 ↔ 𝐴 = ∅) | |
7 | 6 | orbi2i 909 | . . 3 ⊢ ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (∅ ∈ 𝐴 ∨ 𝐴 = ∅)) |
8 | orcom 866 | . . 3 ⊢ ((∅ ∈ 𝐴 ∨ 𝐴 = ∅) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) | |
9 | 7, 8 | bitri 277 | . 2 ⊢ ((∅ ∈ 𝐴 ∨ ∅ = 𝐴) ↔ (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) |
10 | 5, 9 | sylib 220 | 1 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ⊆ wss 3936 ∅c0 4291 Oncon0 6186 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-tr 5166 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-ord 6189 df-on 6190 |
This theorem is referenced by: snsn0non 6304 onxpdisj 6305 omabs 8268 cnfcom3lem 9160 |
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