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Mirrors > Home > MPE Home > Th. List > ordon | Structured version Visualization version GIF version |
Description: The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. (Contributed by NM, 17-May-1994.) |
Ref | Expression |
---|---|
ordon | ⊢ Ord On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tron 5784 | . 2 ⊢ Tr On | |
2 | onfr 5801 | . . 3 ⊢ E Fr On | |
3 | eloni 5771 | . . . . 5 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
4 | eloni 5771 | . . . . 5 ⊢ (𝑦 ∈ On → Ord 𝑦) | |
5 | ordtri3or 5793 | . . . . . 6 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | |
6 | epel 5061 | . . . . . . 7 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
7 | biid 251 | . . . . . . 7 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦) | |
8 | epel 5061 | . . . . . . 7 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
9 | 6, 7, 8 | 3orbi123i 1271 | . . . . . 6 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) |
10 | 5, 9 | sylibr 224 | . . . . 5 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)) |
11 | 3, 4, 10 | syl2an 493 | . . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)) |
12 | 11 | rgen2a 3006 | . . 3 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) |
13 | dfwe2 7023 | . . 3 ⊢ ( E We On ↔ ( E Fr On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))) | |
14 | 2, 12, 13 | mpbir2an 975 | . 2 ⊢ E We On |
15 | df-ord 5764 | . 2 ⊢ (Ord On ↔ (Tr On ∧ E We On)) | |
16 | 1, 14, 15 | mpbir2an 975 | 1 ⊢ Ord On |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∨ w3o 1053 ∈ wcel 2030 ∀wral 2941 class class class wbr 4685 Tr wtr 4785 E cep 5057 Fr wfr 5099 We wwe 5101 Ord word 5760 Oncon0 5761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-tr 4786 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-ord 5764 df-on 5765 |
This theorem is referenced by: epweon 7025 onprc 7026 ssorduni 7027 ordeleqon 7030 ordsson 7031 onint 7037 suceloni 7055 limon 7078 tfi 7095 ordom 7116 ordtypelem2 8465 hartogs 8490 card2on 8500 tskwe 8814 alephsmo 8963 ondomon 9423 dford3lem2 37911 dford3 37912 iunord 42747 |
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