ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ordom GIF version

Theorem ordom 4618
Description: Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.)
Assertion
Ref Expression
ordom Ord ω

Proof of Theorem ordom
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnn 4617 . . . 4 ((𝑥𝑦𝑦 ∈ ω) → 𝑥 ∈ ω)
21gen2 1460 . . 3 𝑥𝑦((𝑥𝑦𝑦 ∈ ω) → 𝑥 ∈ ω)
3 dftr2 4115 . . 3 (Tr ω ↔ ∀𝑥𝑦((𝑥𝑦𝑦 ∈ ω) → 𝑥 ∈ ω))
42, 3mpbir 146 . 2 Tr ω
5 treq 4119 . . . 4 (𝑦 = ∅ → (Tr 𝑦 ↔ Tr ∅))
6 treq 4119 . . . 4 (𝑦 = 𝑥 → (Tr 𝑦 ↔ Tr 𝑥))
7 treq 4119 . . . 4 (𝑦 = suc 𝑥 → (Tr 𝑦 ↔ Tr suc 𝑥))
8 tr0 4124 . . . 4 Tr ∅
9 suctr 4433 . . . . 5 (Tr 𝑥 → Tr suc 𝑥)
109a1i 9 . . . 4 (𝑥 ∈ ω → (Tr 𝑥 → Tr suc 𝑥))
115, 6, 7, 6, 8, 10finds 4611 . . 3 (𝑥 ∈ ω → Tr 𝑥)
1211rgen 2540 . 2 𝑥 ∈ ω Tr 𝑥
13 dford3 4379 . 2 (Ord ω ↔ (Tr ω ∧ ∀𝑥 ∈ ω Tr 𝑥))
144, 12, 13mpbir2an 943 1 Ord ω
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1361  wcel 2158  wral 2465  c0 3434  Tr wtr 4113  Ord word 4374  suc csuc 4377  ωcom 4601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-uni 3822  df-int 3857  df-tr 4114  df-iord 4378  df-suc 4383  df-iom 4602
This theorem is referenced by:  omelon2  4619  limom  4625  freccllem  6417  frecfcllem  6419  frecsuclem  6421  fict  6882  infnfi  6909  isinfinf  6911  hashinfuni  10771  hashinfom  10772  hashennn  10774  ennnfonelemrn  12434  ctinf  12445
  Copyright terms: Public domain W3C validator