Theorem sorpssi 7727

Description: Property of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Assertion

Ref	Expression
sorpssi	⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵))

Proof of Theorem sorpssi

Step	Hyp	Ref	Expression
1		solin 5597	. . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 [⊊] 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 [⊊] 𝐵))
2		elex 3484	. . . . . 6 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V)
3	2	ad2antll 741	. . . . 5 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐶 ∈ V)
4		brrpssg 7723	. . . . 5 ⊢ (𝐶 ∈ V → (𝐵 [⊊] 𝐶 ↔ 𝐵 ⊊ 𝐶))
5	3, 4	syl 18	. . . 4 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 [⊊] 𝐶 ↔ 𝐵 ⊊ 𝐶))
6		biidd 265	. . . 4 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 = 𝐶 ↔ 𝐵 = 𝐶))
7		elex 3484	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V)
8	7	ad2antrl 740	. . . . 5 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ V)
9		brrpssg 7723	. . . . 5 ⊢ (𝐵 ∈ V → (𝐶 [⊊] 𝐵 ↔ 𝐶 ⊊ 𝐵))
10	8, 9	syl 18	. . . 4 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐶 [⊊] 𝐵 ↔ 𝐶 ⊊ 𝐵))
11	5, 6, 10	3orbi123d 1461	. . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝐵 [⊊] 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 [⊊] 𝐵) ↔ (𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ⊊ 𝐵)))
12	1, 11	mpbid 235	. 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ⊊ 𝐵))
13		sspsstri 4068	. 2 ⊢ ((𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵) ↔ (𝐵 ⊊ 𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶 ⊊ 𝐵))
14	12, 13	sylibr 237	1 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∨ w3o 1100   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ⊆ wss 3913   ⊊ wpss 3914   class class class wbr 5113   Or wor 5569   [⊊] crpss 7720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-so 5571  df-xp 5668  df-rel 5669  df-rpss 7721

This theorem is referenced by:  sorpssun  7728  sorpssin  7729  sorpssuni  7730  sorpssint  7731  sorpsscmpl  7732  enfin2i  10305  fin1a2lem9  10392  fin1a2lem10  10393  fin1a2lem11  10394  fin1a2lem13  10396  ssdifidllem  21453  ssmxidllem  33701