MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  onomeneq Structured version   Visualization version   GIF version

Theorem onomeneq 9179
Description: An ordinal number equinumerous to a natural number is equal to it. Proposition 10.22 of [TakeutiZaring] p. 90 and its converse. (Contributed by NM, 26-Jul-2004.) Avoid ax-pow 5325. (Revised by BTernaryTau, 2-Dec-2024.)
Assertion
Ref Expression
onomeneq ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))

Proof of Theorem onomeneq
StepHypRef Expression
1 endom 8926 . . . . . 6 (𝐴𝐵𝐴𝐵)
2 nnfi 9118 . . . . . . . . 9 (𝐵 ∈ ω → 𝐵 ∈ Fin)
3 domfi 9143 . . . . . . . . . . 11 ((𝐵 ∈ Fin ∧ 𝐴𝐵) → 𝐴 ∈ Fin)
4 simpr 486 . . . . . . . . . . 11 ((𝐵 ∈ Fin ∧ 𝐴𝐵) → 𝐴𝐵)
53, 4jca 513 . . . . . . . . . 10 ((𝐵 ∈ Fin ∧ 𝐴𝐵) → (𝐴 ∈ Fin ∧ 𝐴𝐵))
6 domnsymfi 9154 . . . . . . . . . . . . . 14 ((𝐴 ∈ Fin ∧ 𝐴𝐵) → ¬ 𝐵𝐴)
76ex 414 . . . . . . . . . . . . 13 (𝐴 ∈ Fin → (𝐴𝐵 → ¬ 𝐵𝐴))
8 php3 9163 . . . . . . . . . . . . . 14 ((𝐴 ∈ Fin ∧ 𝐵𝐴) → 𝐵𝐴)
98ex 414 . . . . . . . . . . . . 13 (𝐴 ∈ Fin → (𝐵𝐴𝐵𝐴))
107, 9nsyld 156 . . . . . . . . . . . 12 (𝐴 ∈ Fin → (𝐴𝐵 → ¬ 𝐵𝐴))
1110adantl 483 . . . . . . . . . . 11 ((𝐵 ∈ ω ∧ 𝐴 ∈ Fin) → (𝐴𝐵 → ¬ 𝐵𝐴))
1211expimpd 455 . . . . . . . . . 10 (𝐵 ∈ ω → ((𝐴 ∈ Fin ∧ 𝐴𝐵) → ¬ 𝐵𝐴))
135, 12syl5 34 . . . . . . . . 9 (𝐵 ∈ ω → ((𝐵 ∈ Fin ∧ 𝐴𝐵) → ¬ 𝐵𝐴))
142, 13mpand 694 . . . . . . . 8 (𝐵 ∈ ω → (𝐴𝐵 → ¬ 𝐵𝐴))
1514adantl 483 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ¬ 𝐵𝐴))
16 eloni 6332 . . . . . . . 8 (𝐴 ∈ On → Ord 𝐴)
17 nnord 7815 . . . . . . . 8 (𝐵 ∈ ω → Ord 𝐵)
18 ordtri1 6355 . . . . . . . . 9 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
19 ordelpss 6350 . . . . . . . . . . 11 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵𝐴𝐵𝐴))
2019ancoms 460 . . . . . . . . . 10 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵𝐴𝐵𝐴))
2120notbid 318 . . . . . . . . 9 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐵𝐴 ↔ ¬ 𝐵𝐴))
2218, 21bitrd 279 . . . . . . . 8 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
2316, 17, 22syl2an 597 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
2415, 23sylibrd 259 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴𝐵))
251, 24syl5 34 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴𝐵))
26253impia 1118 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐴𝐵)
27 ensymfib 9138 . . . . . . . . 9 (𝐵 ∈ Fin → (𝐵𝐴𝐴𝐵))
282, 27syl 17 . . . . . . . 8 (𝐵 ∈ ω → (𝐵𝐴𝐴𝐵))
29 endom 8926 . . . . . . . 8 (𝐵𝐴𝐵𝐴)
3028, 29syl6bir 254 . . . . . . 7 (𝐵 ∈ ω → (𝐴𝐵𝐵𝐴))
3130imp 408 . . . . . 6 ((𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐵𝐴)
32313adant1 1131 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐵𝐴)
33 nndomog 9167 . . . . . . 7 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (𝐵𝐴𝐵𝐴))
3433ancoms 460 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐵𝐴𝐵𝐴))
3534biimp3a 1470 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ ω ∧ 𝐵𝐴) → 𝐵𝐴)
3632, 35syld3an3 1410 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐵𝐴)
3726, 36eqssd 3966 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐴 = 𝐵)
38373expia 1122 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
39 enrefnn 8998 . . . 4 (𝐵 ∈ ω → 𝐵𝐵)
40 breq1 5113 . . . 4 (𝐴 = 𝐵 → (𝐴𝐵𝐵𝐵))
4139, 40syl5ibrcom 247 . . 3 (𝐵 ∈ ω → (𝐴 = 𝐵𝐴𝐵))
4241adantl 483 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵𝐴𝐵))
4338, 42impbid 211 1 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wss 3915  wpss 3916   class class class wbr 5110  Ord word 6321  Oncon0 6322  ωcom 7807  cen 8887  cdom 8888  csdm 8889  Fincfn 8890
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
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-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-om 7808  df-1o 8417  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894
This theorem is referenced by:  onfin  9181  ficardom  9904  finnisoeu  10056
  Copyright terms: Public domain W3C validator