Theorem sorpss 7701

Description: Express strict ordering under proper subsets, i.e. the notion of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Assertion

Ref	Expression
sorpss	⊢ ( [⊊] Or 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem sorpss

Step	Hyp	Ref	Expression
1		porpss 7700	. . 3 ⊢ [⊊] Po 𝐴
2	1	biantrur 531	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥) ↔ ( [⊊] Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥)))
3		sspsstri 4098	. . . 4 ⊢ ((𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ (𝑥 ⊊ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ⊊ 𝑥))
4		vex 3477	. . . . . 6 ⊢ 𝑦 ∈ V
5	4	brrpss 7699	. . . . 5 ⊢ (𝑥 [⊊] 𝑦 ↔ 𝑥 ⊊ 𝑦)
6		biid 260	. . . . 5 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
7		vex 3477	. . . . . 6 ⊢ 𝑥 ∈ V
8	7	brrpss 7699	. . . . 5 ⊢ (𝑦 [⊊] 𝑥 ↔ 𝑦 ⊊ 𝑥)
9	5, 6, 8	3orbi123i 1156	. . . 4 ⊢ ((𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥) ↔ (𝑥 ⊊ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ⊊ 𝑥))
10	3, 9	bitr4i 277	. . 3 ⊢ ((𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ (𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥))
11	10	2ralbii 3127	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥))
12		df-so 5582	. 2 ⊢ ( [⊊] Or 𝐴 ↔ ( [⊊] Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥)))
13	2, 11, 12	3bitr4ri 303	1 ⊢ ( [⊊] Or 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∨ w3o 1086  ∀wral 3060   ⊆ wss 3944   ⊊ wpss 3945   class class class wbr 5141   Po wpo 5579   Or wor 5580   [⊊] crpss 7695

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-po 5581  df-so 5582  df-xp 5675  df-rel 5676  df-rpss 7696

This theorem is referenced by:  sorpsscmpl  7707  enfin2i  10298  fin1a2lem13  10389