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| Mirrors > Home > MPE Home > Th. List > orddif | Structured version Visualization version GIF version | ||
| Description: Ordinal derived from its successor. (Contributed by NM, 20-May-1998.) |
| Ref | Expression |
|---|---|
| orddif | ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | orddisj 6348 | . 2 ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) | |
| 2 | disj3 4382 | . . 3 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (𝐴 ∖ {𝐴})) | |
| 3 | df-suc 6316 | . . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
| 4 | 3 | difeq1i 4053 | . . . . 5 ⊢ (suc 𝐴 ∖ {𝐴}) = ((𝐴 ∪ {𝐴}) ∖ {𝐴}) |
| 5 | difun2 4409 | . . . . 5 ⊢ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = (𝐴 ∖ {𝐴}) | |
| 6 | 4, 5 | eqtri 2762 | . . . 4 ⊢ (suc 𝐴 ∖ {𝐴}) = (𝐴 ∖ {𝐴}) |
| 7 | 6 | eqeq2i 2752 | . . 3 ⊢ (𝐴 = (suc 𝐴 ∖ {𝐴}) ↔ 𝐴 = (𝐴 ∖ {𝐴})) |
| 8 | 2, 7 | bitr4i 279 | . 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ 𝐴 = (suc 𝐴 ∖ {𝐴})) |
| 9 | 1, 8 | sylib 219 | 1 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1547 ∖ cdif 3880 ∪ cun 3881 ∩ cin 3882 ∅c0 4261 {csn 4555 Ord word 6309 suc csuc 6312 |
| 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 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-ext 2711 ax-sep 5218 ax-pr 5362 |
| This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-sb 2074 df-clab 2718 df-cleq 2731 df-clel 2814 df-ne 2935 df-ral 3054 df-rex 3064 df-rab 3392 df-v 3433 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-br 5073 df-opab 5135 df-eprel 5518 df-fr 5571 df-we 5573 df-ord 6313 df-suc 6316 |
| This theorem is referenced by: dif1enlem 9084 pssnn 9093 phplem2 9129 cantnfresb 43769 |
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