Theorem cofcutrtime1d 28315

Description: If 𝑋 is a timely cut of 𝐴 and 𝐵, then (L‘𝑋) is cofinal with 𝐴. (Contributed by Scott Fenton, 23-Jan-2025.)

Hypotheses

Ref	Expression
cofcutrtimed.1	⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)))
cofcutrtimed.2	⊢ (𝜑 → 𝐴 <<s 𝐵)
cofcutrtimed.3	⊢ (𝜑 → 𝑋 = (𝐴 |s 𝐵))

Assertion

Ref	Expression
cofcutrtime1d	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (L‘𝑋)𝑥 ≤s 𝑦)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑋,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐴(𝑦)   𝐵(𝑦)

Proof of Theorem cofcutrtime1d

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cofcutrtimed.1	. . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)))
2		cofcutrtimed.2	. . 3 ⊢ (𝜑 → 𝐴 <<s 𝐵)
3		cofcutrtimed.3	. . 3 ⊢ (𝜑 → 𝑋 = (𝐴 |s 𝐵))
4		cofcutrtime 28314	. . 3 ⊢ (((𝐴 ∪ 𝐵) ⊆ (O‘(bday‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (L‘𝑋)𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ (R‘𝑋)𝑤 ≤s 𝑧))
5	1, 2, 3, 4	syl3anc 1398	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (L‘𝑋)𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ (R‘𝑋)𝑤 ≤s 𝑧))
6	5	simpld 500	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (L‘𝑋)𝑥 ≤s 𝑦)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∀wral 3077  ∃wrex 3087   ∪ cun 3897   ⊆ wss 3914   class class class wbr 5103  ‘cfv 6539  (class class class)co 7421  bdaycbday 28000   ≤s cles 28102   <<s cslts 28144   |s ccuts 28146  Ocold 28210  Lcleft 28212  Rcright 28213

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3900  df-un 3902  df-in 3904  df-ss 3915  df-pss 4035  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6305  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-2nd 8003  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-1o 8477  df-2o 8478  df-no 28001  df-lts 28002  df-bday 28003  df-les 28103  df-slts 28145  df-cuts 28147  df-made 28214  df-old 28215  df-left 28217  df-right 28218

This theorem is used by: (None)