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Theorem coflton 8665
Description: Cofinality theorem for ordinals. If 𝐴 is cofinal with 𝐵 and 𝐵 precedes 𝐶, then 𝐴 precedes 𝐶. Compare cofsslt 27742 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 3999 . . . . . . 7 (𝑥 = 𝑎 → (𝑥𝑦𝑎𝑦))
21rexbidv 3170 . . . . . 6 (𝑥 = 𝑎 → (∃𝑦𝐵 𝑥𝑦 ↔ ∃𝑦𝐵 𝑎𝑦))
3 coflton.4 . . . . . . 7 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
43adantr 480 . . . . . 6 ((𝜑𝑎𝐴) → ∀𝑥𝐴𝑦𝐵 𝑥𝑦)
5 simpr 484 . . . . . 6 ((𝜑𝑎𝐴) → 𝑎𝐴)
62, 4, 5rspcdva 3605 . . . . 5 ((𝜑𝑎𝐴) → ∃𝑦𝐵 𝑎𝑦)
76adantrr 714 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑐𝐶)) → ∃𝑦𝐵 𝑎𝑦)
8 sseq2 4000 . . . . 5 (𝑦 = 𝑏 → (𝑎𝑦𝑎𝑏))
98cbvrexvw 3227 . . . 4 (∃𝑦𝐵 𝑎𝑦 ↔ ∃𝑏𝐵 𝑎𝑏)
107, 9sylib 217 . . 3 ((𝜑 ∧ (𝑎𝐴𝑐𝐶)) → ∃𝑏𝐵 𝑎𝑏)
11 simpr 484 . . . . . 6 (((𝜑 ∧ (𝑎𝐴𝑐𝐶)) ∧ 𝑏𝐵) → 𝑏𝐵)
12 simplrr 775 . . . . . 6 (((𝜑 ∧ (𝑎𝐴𝑐𝐶)) ∧ 𝑏𝐵) → 𝑐𝐶)
13 coflton.5 . . . . . . 7 (𝜑 → ∀𝑧𝐵𝑤𝐶 𝑧𝑤)
1413ad2antrr 723 . . . . . 6 (((𝜑 ∧ (𝑎𝐴𝑐𝐶)) ∧ 𝑏𝐵) → ∀𝑧𝐵𝑤𝐶 𝑧𝑤)
15 elequ1 2105 . . . . . . 7 (𝑧 = 𝑏 → (𝑧𝑤𝑏𝑤))
16 elequ2 2113 . . . . . . 7 (𝑤 = 𝑐 → (𝑏𝑤𝑏𝑐))
1715, 16rspc2va 3615 . . . . . 6 (((𝑏𝐵𝑐𝐶) ∧ ∀𝑧𝐵𝑤𝐶 𝑧𝑤) → 𝑏𝑐)
1811, 12, 14, 17syl21anc 835 . . . . 5 (((𝜑 ∧ (𝑎𝐴𝑐𝐶)) ∧ 𝑏𝐵) → 𝑏𝑐)
19 coflton.1 . . . . . . . 8 (𝜑𝐴 ⊆ On)
2019sselda 3974 . . . . . . 7 ((𝜑𝑎𝐴) → 𝑎 ∈ On)
2120adantrr 714 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑐𝐶)) → 𝑎 ∈ On)
22 coflton.3 . . . . . . . . 9 (𝜑𝐶 ⊆ On)
2322sselda 3974 . . . . . . . 8 ((𝜑𝑐𝐶) → 𝑐 ∈ On)
2423adantrl 713 . . . . . . 7 ((𝜑 ∧ (𝑎𝐴𝑐𝐶)) → 𝑐 ∈ On)
2524adantr 480 . . . . . 6 (((𝜑 ∧ (𝑎𝐴𝑐𝐶)) ∧ 𝑏𝐵) → 𝑐 ∈ On)
26 ontr2 6401 . . . . . 6 ((𝑎 ∈ On ∧ 𝑐 ∈ On) → ((𝑎𝑏𝑏𝑐) → 𝑎𝑐))
2721, 25, 26syl2an2r 682 . . . . 5 (((𝜑 ∧ (𝑎𝐴𝑐𝐶)) ∧ 𝑏𝐵) → ((𝑎𝑏𝑏𝑐) → 𝑎𝑐))
2818, 27mpan2d 691 . . . 4 (((𝜑 ∧ (𝑎𝐴𝑐𝐶)) ∧ 𝑏𝐵) → (𝑎𝑏𝑎𝑐))
2928rexlimdva 3147 . . 3 ((𝜑 ∧ (𝑎𝐴𝑐𝐶)) → (∃𝑏𝐵 𝑎𝑏𝑎𝑐))
3010, 29mpd 15 . 2 ((𝜑 ∧ (𝑎𝐴𝑐𝐶)) → 𝑎𝑐)
3130ralrimivva 3192 1 (𝜑 → ∀𝑎𝐴𝑐𝐶 𝑎𝑐)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098  wral 3053  wrex 3062  wss 3940  Oncon0 6354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-tr 5256  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-ord 6357  df-on 6358
This theorem is referenced by: (None)
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