Theorem sorpss 7704

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 7703	. . 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 4068	. . . 4 ⊢ ((𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ (𝑥 ⊊ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ⊊ 𝑥))
4		vex 3451	. . . . . 6 ⊢ 𝑦 ∈ V
5	4	brrpss 7702	. . . . 5 ⊢ (𝑥 [⊊] 𝑦 ↔ 𝑥 ⊊ 𝑦)
6		biid 261	. . . . 5 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
7		vex 3451	. . . . . 6 ⊢ 𝑥 ∈ V
8	7	brrpss 7702	. . . . 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 278	. . 3 ⊢ ((𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ (𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥))
11	10	2ralbii 3108	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 [⊊] 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 [⊊] 𝑥))
12		df-so 5547	. 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 847   ∨ w3o 1085  ∀wral 3044   ⊆ wss 3914   ⊊ wpss 3915   class class class wbr 5107   Po wpo 5544   Or wor 5545   [⊊] crpss 7698

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-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-po 5546  df-so 5547  df-xp 5644  df-rel 5645  df-rpss 7699

This theorem is referenced by:  sorpsscmpl  7710  enfin2i  10274  fin1a2lem13  10365