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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 5663 | . 2 ⊢ Tr On | |
2 | onfr 5680 | . . 3 ⊢ E Fr On | |
3 | eloni 5650 | . . . . 5 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
4 | eloni 5650 | . . . . 5 ⊢ (𝑦 ∈ On → Ord 𝑦) | |
5 | ordtri3or 5672 | . . . . . 6 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | |
6 | epel 4952 | . . . . . . 7 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
7 | biid 250 | . . . . . . 7 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦) | |
8 | epel 4952 | . . . . . . 7 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
9 | 6, 7, 8 | 3orbi123i 1245 | . . . . . 6 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) |
10 | 5, 9 | sylibr 223 | . . . . 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 2960 | . . 3 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) |
13 | dfwe2 6873 | . . 3 ⊢ ( E We On ↔ ( E Fr On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))) | |
14 | 2, 12, 13 | mpbir2an 957 | . 2 ⊢ E We On |
15 | df-ord 5643 | . 2 ⊢ (Ord On ↔ (Tr On ∧ E We On)) | |
16 | 1, 14, 15 | mpbir2an 957 | 1 ⊢ Ord On |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∨ w3o 1030 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 Tr wtr 4680 E cep 4947 Fr wfr 4994 We wwe 4996 Ord word 5639 Oncon0 5640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 |
This theorem is referenced by: epweon 6875 onprc 6876 ssorduni 6877 ordeleqon 6880 ordsson 6881 onint 6887 suceloni 6905 limon 6928 tfi 6945 ordom 6966 ordtypelem2 8307 hartogs 8332 card2on 8342 tskwe 8659 alephsmo 8808 ondomon 9264 dford3lem2 36612 dford3 36613 iunord 42220 |
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