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Mirrors > Home > MPE Home > Th. List > nordeq | Structured version Visualization version GIF version |
Description: A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.) |
Ref | Expression |
---|---|
nordeq | ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordirr 6382 | . . . 4 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
2 | eleq1 2821 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
3 | 2 | notbid 317 | . . . 4 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 ∈ 𝐴 ↔ ¬ 𝐵 ∈ 𝐴)) |
4 | 1, 3 | syl5ibcom 244 | . . 3 ⊢ (Ord 𝐴 → (𝐴 = 𝐵 → ¬ 𝐵 ∈ 𝐴)) |
5 | 4 | necon2ad 2955 | . 2 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → 𝐴 ≠ 𝐵)) |
6 | 5 | imp 407 | 1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 Ord word 6363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-eprel 5580 df-fr 5631 df-we 5633 df-ord 6367 |
This theorem is referenced by: php 9212 phplem1OLD 9219 phpOLD 9224 nogt01o 27423 ordtop 35624 limsucncmpi 35633 |
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