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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 8318 | . 2 ⊢ (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ 𝐴 ≠ ∅)) | |
2 | on0eln0 6315 | . . 3 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | |
3 | 2 | pm5.32i 575 | . 2 ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) ↔ (𝐴 ∈ On ∧ 𝐴 ≠ ∅)) |
4 | 1, 3 | bitr4i 277 | 1 ⊢ (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3884 ∅c0 4257 Oncon0 6260 1oc1o 8278 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-11 2154 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pr 5351 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3432 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-br 5075 df-opab 5137 df-tr 5192 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-ord 6263 df-on 6264 df-suc 6266 df-1o 8285 |
This theorem is referenced by: cantnflem2 9436 oef1o 9444 cnfcom3 9450 infxpenc 9762 |
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