Theorem sorpss 7683

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 7682	. . 3 ⊢ [⊊] Po 𝐴
2	1	biantrur 530	. 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 4059	. . . 4 ⊢ ((𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ (𝑥 ⊊ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ⊊ 𝑥))
4		vex 3446	. . . . . 6 ⊢ 𝑦 ∈ V
5	4	brrpss 7681	. . . . 5 ⊢ (𝑥 [⊊] 𝑦 ↔ 𝑥 ⊊ 𝑦)
6		biid 261	. . . . 5 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
7		vex 3446	. . . . . 6 ⊢ 𝑥 ∈ V
8	7	brrpss 7681	. . . . 5 ⊢ (𝑦 [⊊] 𝑥 ↔ 𝑦 ⊊ 𝑥)
9	5, 6, 8	3orbi123i 1157	. . . 4 ⊢ ((𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥) ↔ (𝑥 ⊊ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ⊊ 𝑥))
10	3, 9	bitr4i 278	. . 3 ⊢ ((𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ (𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥))
11	10	2ralbii 3113	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥))
12		df-so 5541	. 2 ⊢ ( [⊊] Or 𝐴 ↔ ( [⊊] Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥)))
13	2, 11, 12	3bitr4ri 304	1 ⊢ ( [⊊] Or 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∨ w3o 1086  ∀wral 3052   ⊆ wss 3903   ⊊ wpss 3904   class class class wbr 5100   Po wpo 5538   Or wor 5539   [⊊] crpss 7677

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-ext 2709  ax-sep 5243  ax-pr 5379

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-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-po 5540  df-so 5541  df-xp 5638  df-rel 5639  df-rpss 7678

This theorem is referenced by:  sorpsscmpl  7689  enfin2i  10243  fin1a2lem13  10334