Theorem frirr 5496

Description: A well-founded relation is irreflexive. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 2-Jan-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)

Assertion

Ref	Expression
frirr	⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)

Proof of Theorem frirr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 486	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝑅 Fr 𝐴)
2		snssi 4701	. . . 4 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴)
3	2	adantl 485	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝐵} ⊆ 𝐴)
4		snnzg 4670	. . . 4 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ≠ ∅)
5	4	adantl 485	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝐵} ≠ ∅)
6		snex 5297	. . . 4 ⊢ {𝐵} ∈ V
7	6	frc 5485	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ {𝐵} ⊆ 𝐴 ∧ {𝐵} ≠ ∅) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅)
8	1, 3, 5, 7	syl3anc 1368	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅)
9		rabeq0 4292	. . . . . 6 ⊢ ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝑦)
10		breq2 5034	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐵))
11	10	notbid 321	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐵))
12	11	ralbidv 3162	. . . . . 6 ⊢ (𝑦 = 𝐵 → (∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵))
13	9, 12	syl5bb 286	. . . . 5 ⊢ (𝑦 = 𝐵 → ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵))
14	13	rexsng 4574	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵))
15		breq1 5033	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝐵 ↔ 𝐵𝑅𝐵))
16	15	notbid 321	. . . . 5 ⊢ (𝑥 = 𝐵 → (¬ 𝑥𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
17	16	ralsng 4573	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
18	14, 17	bitrd 282	. . 3 ⊢ (𝐵 ∈ 𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵))
19	18	adantl 485	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵))
20	8, 19	mpbid 235	1 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110   ⊆ wss 3881  ∅c0 4243  {csn 4525   class class class wbr 5030   Fr wfr 5475

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-fr 5478

This theorem is referenced by:  efrirr  5500  predfrirr  6145  dfwe2  7476  bnj1417  32423  efrunt  33052  ifr0  41154