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Mirrors > Home > MPE Home > Th. List > onirri | Structured version Visualization version GIF version |
Description: An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.) |
Ref | Expression |
---|---|
on.1 | ⊢ 𝐴 ∈ On |
Ref | Expression |
---|---|
onirri | ⊢ ¬ 𝐴 ∈ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | on.1 | . . 3 ⊢ 𝐴 ∈ On | |
2 | 1 | onordi 6044 | . 2 ⊢ Ord 𝐴 |
3 | ordirr 5958 | . 2 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ ¬ 𝐴 ∈ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2157 Ord word 5939 Oncon0 5940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-sep 4974 ax-nul 4982 ax-pr 5096 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3386 df-sbc 3633 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-nul 4115 df-if 4277 df-sn 4368 df-pr 4370 df-op 4374 df-uni 4628 df-br 4843 df-opab 4905 df-tr 4945 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-we 5272 df-ord 5943 df-on 5944 |
This theorem is referenced by: onssnel2i 6050 onuninsuci 7273 oelim2 7914 omopthlem2 7975 harndom 8710 wfelirr 8937 carduni 9092 pm54.43 9111 alephle 9196 alephfp 9216 pwxpndom2 9774 onsucsuccmpi 32943 onint1 32949 finxpreclem5 33723 wepwsolem 38386 |
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