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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 6290 | . 2 ⊢ Ord 𝐴 |
3 | ordirr 6204 | . 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 2110 Ord word 6185 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-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-nul 4292 df-if 4468 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: onssnel2i 6296 onuninsuci 7549 oelim2 8215 omopthlem2 8277 enpr2d 8591 harndom 9022 wfelirr 9248 carduni 9404 pm54.43 9423 alephle 9508 alephfp 9528 pwxpndom2 10081 onsucsuccmpi 33786 onint1 33792 finxpreclem5 34670 wepwsolem 39635 |
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