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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 6495 | . 2 ⊢ Ord 𝐴 |
| 3 | ordirr 6402 | . 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 2108 Ord word 6383 Oncon0 6384 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 |
| This theorem is referenced by: onssnel2i 6501 onuninsuci 7861 nlim2 8528 ord1eln01 8534 ord2eln012 8535 oelim2 8633 omopthlem2 8698 enpr2dOLD 9090 harndom 9602 ssttrcl 9755 wfelirr 9865 carduni 10021 pm54.43 10041 alephle 10128 alephfp 10148 pwxpndom2 10705 oldirr 27928 lrrecpo 27974 onsucsuccmpi 36444 onint1 36450 finxpreclem5 37396 wepwsolem 43054 |
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