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Mirrors > Home > ILE Home > Th. List > ordn2lp | GIF version |
Description: An ordinal class cannot be an element of one of its members. Variant of first part of Theorem 2.2(vii) of [BellMachover] p. 469. (Contributed by NM, 3-Apr-1994.) |
Ref | Expression |
---|---|
ordn2lp | ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordirr 4371 | . 2 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
2 | ordtr 4214 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
3 | trel 3949 | . . 3 ⊢ (Tr 𝐴 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝐴 ∈ 𝐴)) | |
4 | 2, 3 | syl 14 | . 2 ⊢ (Ord 𝐴 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝐴 ∈ 𝐴)) |
5 | 1, 4 | mtod 625 | 1 ⊢ (Ord 𝐴 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∈ wcel 1439 Tr wtr 3942 Ord word 4198 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-setind 4366 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-v 2622 df-dif 3002 df-in 3006 df-ss 3013 df-sn 3456 df-uni 3660 df-tr 3943 df-iord 4202 |
This theorem is referenced by: nninfalllemn 12170 |
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