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Mirrors > Home > MPE Home > Th. List > onsucmin | Structured version Visualization version GIF version |
Description: The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.) |
Ref | Expression |
---|---|
onsucmin | ⊢ (𝐴 ∈ On → suc 𝐴 = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 6375 | . . . . 5 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
2 | ordelsuc 7808 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ Ord 𝑥) → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ⊆ 𝑥)) | |
3 | 1, 2 | sylan2 594 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ⊆ 𝑥)) |
4 | 3 | rabbidva 3440 | . . 3 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴 ∈ 𝑥} = {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥}) |
5 | 4 | inteqd 4956 | . 2 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ 𝐴 ∈ 𝑥} = ∩ {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥}) |
6 | onsucb 7805 | . . 3 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) | |
7 | intmin 4973 | . . 3 ⊢ (suc 𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥} = suc 𝐴) | |
8 | 6, 7 | sylbi 216 | . 2 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥} = suc 𝐴) |
9 | 5, 8 | eqtr2d 2774 | 1 ⊢ (𝐴 ∈ On → suc 𝐴 = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 {crab 3433 ⊆ wss 3949 ∩ cint 4951 Ord word 6364 Oncon0 6365 suc csuc 6367 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-br 5150 df-opab 5212 df-tr 5267 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-ord 6368 df-on 6369 df-suc 6371 |
This theorem is referenced by: ranksnb 9822 nadd1suc 42142 naddsuc2 42143 |
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