Theorem frege111d 44207

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 44422. (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 44204	. 2 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵))
8	7	frege114d 44206	1 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝑅)𝐴))

Syntax hints:   → wi 4   ∨ wo 848   ∨ w3o 1086   = wceq 1542   ∈ wcel 2114  Vcvv 3430   class class class wbr 5086  ‘cfv 6493  t+ctcl 14941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-iota 6449  df-fun 6495  df-fv 6501  df-trcl 14943

This theorem is referenced by: (None)