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

Theorem coflton 8618
Description: Cofinality theorem for ordinals. If 𝐴 is cofinal with 𝐵 and 𝐵 precedes 𝐶, then 𝐴 precedes 𝐶. Compare cofsslt 27240 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
Hypotheses
Ref Expression
coflton.1 (𝜑𝐴 ⊆ On)
coflton.2 (𝜑𝐵 ⊆ On)
coflton.3 (𝜑𝐶 ⊆ On)
coflton.4 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
coflton.5 (𝜑 → ∀𝑧𝐵𝑤𝐶 𝑧𝑤)
Assertion
Ref Expression
coflton (𝜑 → ∀𝑎𝐴𝑐𝐶 𝑎𝑐)
Distinct variable groups:   𝐴,𝑐   𝑥,𝐴   𝑥,𝐵,𝑦   𝑧,𝐵   𝑤,𝐶,𝑧   𝑎,𝑐,𝜑   𝑥,𝑎,𝑦   𝑤,𝑐
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝐴(𝑦,𝑧,𝑤,𝑎)   𝐵(𝑤,𝑎,𝑐)   𝐶(𝑥,𝑦,𝑎,𝑐)

Proof of Theorem coflton
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 sseq1 3970 . . . . . . 7 (𝑥 = 𝑎 → (𝑥𝑦𝑎𝑦))
21rexbidv 3176 . . . . . 6 (𝑥 = 𝑎 → (∃𝑦𝐵 𝑥𝑦 ↔ ∃𝑦𝐵 𝑎𝑦))
3 coflton.4 . . . . . . 7 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
43adantr 482 . . . . . 6 ((𝜑𝑎𝐴) → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
5 simpr 486 . . . . . 6 ((𝜑𝑎𝐴) → 𝑎𝐴)
62, 4, 5rspcdva 3583 . . . . 5 ((𝜑𝑎𝐴) → ∃𝑦𝐵 𝑎𝑦)
76adantrr 716 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑐𝐶)) → ∃𝑦𝐵 𝑎𝑦)
8 sseq2 3971 . . . . 5 (𝑦 = 𝑏 → (𝑎𝑦𝑎𝑏))
98cbvrexvw 3227 . . . 4 (∃𝑦𝐵 𝑎𝑦 ↔ ∃𝑏𝐵 𝑎𝑏)
107, 9sylib 217 . . 3 ((𝜑 ∧ (𝑎𝐴𝑐𝐶)) → ∃𝑏𝐵 𝑎𝑏)
11 simpr 486 . . . . . 6 (((𝜑 ∧ (𝑎𝐴𝑐𝐶)) ∧ 𝑏𝐵) → 𝑏𝐵)
12 simplrr 777 . . . . . 6 (((𝜑 ∧ (𝑎𝐴𝑐𝐶)) ∧ 𝑏𝐵) → 𝑐𝐶)
13 coflton.5 . . . . . . 7 (𝜑 → ∀𝑧𝐵𝑤𝐶 𝑧𝑤)
1413ad2antrr 725 . . . . . 6 (((𝜑 ∧ (𝑎𝐴𝑐𝐶)) ∧ 𝑏𝐵) → ∀𝑧𝐵𝑤𝐶 𝑧𝑤)
15 elequ1 2114 . . . . . . 7 (𝑧 = 𝑏 → (𝑧𝑤𝑏𝑤))
16 elequ2 2122 . . . . . . 7 (𝑤 = 𝑐 → (𝑏𝑤𝑏𝑐))
1715, 16rspc2va 3592 . . . . . 6 (((𝑏𝐵𝑐𝐶) ∧ ∀𝑧𝐵𝑤𝐶 𝑧𝑤) → 𝑏𝑐)
1811, 12, 14, 17syl21anc 837 . . . . 5 (((𝜑 ∧ (𝑎𝐴𝑐𝐶)) ∧ 𝑏𝐵) → 𝑏𝑐)
19 coflton.1 . . . . . . . 8 (𝜑𝐴 ⊆ On)
2019sselda 3945 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑎 ∈ On)
2120adantrr 716 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑐𝐶)) → 𝑎 ∈ On)
22 coflton.3 . . . . . . . . 9 (𝜑𝐶 ⊆ On)
2322sselda 3945 . . . . . . . 8 ((𝜑𝑐𝐶) → 𝑐 ∈ On)
2423adantrl 715 . . . . . . 7 ((𝜑 ∧ (𝑎𝐴𝑐𝐶)) → 𝑐 ∈ On)
2524adantr 482 . . . . . 6 (((𝜑 ∧ (𝑎𝐴𝑐𝐶)) ∧ 𝑏𝐵) → 𝑐 ∈ On)
26 ontr2 6365 . . . . . 6 ((𝑎 ∈ On ∧ 𝑐 ∈ On) → ((𝑎𝑏𝑏𝑐) → 𝑎𝑐))
2721, 25, 26syl2an2r 684 . . . . 5 (((𝜑 ∧ (𝑎𝐴𝑐𝐶)) ∧ 𝑏𝐵) → ((𝑎𝑏𝑏𝑐) → 𝑎𝑐))
2818, 27mpan2d 693 . . . 4 (((𝜑 ∧ (𝑎𝐴𝑐𝐶)) ∧ 𝑏𝐵) → (𝑎𝑏𝑎𝑐))
2928rexlimdva 3153 . . 3 ((𝜑 ∧ (𝑎𝐴𝑐𝐶)) → (∃𝑏𝐵 𝑎𝑏𝑎𝑐))
3010, 29mpd 15 . 2 ((𝜑 ∧ (𝑎𝐴𝑐𝐶)) → 𝑎𝑐)
3130ralrimivva 3198 1 (𝜑 → ∀𝑎𝐴𝑐𝐶 𝑎𝑐)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wral 3065  wrex 3074  wss 3911  Oncon0 6318
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-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
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-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-tr 5224  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-ord 6321  df-on 6322
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator