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Mirrors > Home > MPE Home > Th. List > ondif1 | Structured version Visualization version GIF version |
Description: Two ways to say that 𝐴 is a nonzero ordinal number. (Contributed by Mario Carneiro, 21-May-2015.) |
Ref | Expression |
---|---|
ondif1 | ⊢ (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dif1o 8159 | . 2 ⊢ (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ 𝐴 ≠ ∅)) | |
2 | on0eln0 6228 | . . 3 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | |
3 | 2 | pm5.32i 578 | . 2 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ↔ (𝐴 ∈ On ∧ 𝐴 ≠ ∅)) |
4 | 1, 3 | bitr4i 281 | 1 ⊢ (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2114 ≠ wne 2935 ∖ cdif 3841 ∅c0 4212 Oncon0 6173 1oc1o 8127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-11 2162 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pr 5297 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-opab 5094 df-tr 5138 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-ord 6176 df-on 6177 df-suc 6179 df-1o 8134 |
This theorem is referenced by: cantnflem2 9229 oef1o 9237 cnfcom3 9243 infxpenc 9521 |
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