Theorem frege111d 43741

Description: If either 𝐴 and 𝐶 are the same or 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 is the successor to 𝐶, then either 𝐴 and 𝐵 are the same or 𝐴 follows 𝐵 or 𝐵 and 𝐴 in the transitive closure of 𝑅. Similar to Proposition 111 of [Frege1879] p. 75. Compare with frege111 43956. (Contributed by RP, 15-Jul-2020.)

Hypotheses

Ref	Expression
frege111d.r	⊢ (𝜑 → 𝑅 ∈ V)
frege111d.a	⊢ (𝜑 → 𝐴 ∈ V)
frege111d.b	⊢ (𝜑 → 𝐵 ∈ V)
frege111d.c	⊢ (𝜑 → 𝐶 ∈ V)
frege111d.ac	⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶))
frege111d.cb	⊢ (𝜑 → 𝐶𝑅𝐵)

Assertion

Ref	Expression
frege111d	⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝑅)𝐴))

Proof of Theorem frege111d

Step	Hyp	Ref	Expression
1		frege111d.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
2		frege111d.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
3		frege111d.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
4		frege111d.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
5		frege111d.ac	. . 3 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶))
6		frege111d.cb	. . 3 ⊢ (𝜑 → 𝐶𝑅𝐵)
7	1, 2, 3, 4, 5, 6	frege108d 43738	. 2 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵))
8	7	frege114d 43740	1 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝑅)𝐴))

Syntax hints:   → wi 4   ∨ wo 847   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109  Vcvv 3450   class class class wbr 5109  ‘cfv 6513  t+ctcl 14957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4913  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-iota 6466  df-fun 6515  df-fv 6521  df-trcl 14959

This theorem is referenced by: (None)