Theorem frirr 5617

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 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝑅 Fr 𝐴)
2		snssi 4775	. . . 4 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴)
3	2	adantl 481	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝐵} ⊆ 𝐴)
4		snnzg 4741	. . . 4 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ≠ ∅)
5	4	adantl 481	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝐵} ≠ ∅)
6		snex 5394	. . . 4 ⊢ {𝐵} ∈ V
7	6	frc 5604	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ {𝐵} ⊆ 𝐴 ∧ {𝐵} ≠ ∅) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅)
8	1, 3, 5, 7	syl3anc 1373	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅)
9		breq1 5113	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑧𝑅𝑦))
10	9	rabeq0w 4353	. . . . . 6 ⊢ ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝑦)
11		breq2 5114	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝐵))
12	11	notbid 318	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑧𝑅𝐵))
13	12	ralbidv 3157	. . . . . 6 ⊢ (𝑦 = 𝐵 → (∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝑦 ↔ ∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝐵))
14	10, 13	bitrid 283	. . . . 5 ⊢ (𝑦 = 𝐵 → ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝐵))
15	14	rexsng 4643	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝐵))
16		breq1 5113	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧𝑅𝐵 ↔ 𝐵𝑅𝐵))
17	16	notbid 318	. . . . 5 ⊢ (𝑧 = 𝐵 → (¬ 𝑧𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
18	17	ralsng 4642	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (∀𝑧 ∈ {𝐵} ¬ 𝑧𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
19	15, 18	bitrd 279	. . 3 ⊢ (𝐵 ∈ 𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵))
20	19	adantl 481	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵))
21	8, 20	mpbid 232	1 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  {crab 3408   ⊆ wss 3917  ∅c0 4299  {csn 4592   class class class wbr 5110   Fr wfr 5591

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-fr 5594

This theorem is referenced by:  efrirr  5621  predfrirr  6310  dfwe2  7753  bnj1417  35038  efrunt  35707  ifr0  44446