Theorem frirr 5051

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 473	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝑅 Fr 𝐴)
2		snssi 4308	. . . 4 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴)
3	2	adantl 482	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝐵} ⊆ 𝐴)
4		snnzg 4278	. . . 4 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ≠ ∅)
5	4	adantl 482	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝐵} ≠ ∅)
6		snex 4869	. . . 4 ⊢ {𝐵} ∈ V
7	6	frc 5040	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ {𝐵} ⊆ 𝐴 ∧ {𝐵} ≠ ∅) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅)
8	1, 3, 5, 7	syl3anc 1323	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅)
9		rabeq0 3931	. . . . . 6 ⊢ ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝑦)
10		breq2 4617	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐵))
11	10	notbid 308	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝐵))
12	11	ralbidv 2980	. . . . . 6 ⊢ (𝑦 = 𝐵 → (∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝑦 ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵))
13	9, 12	syl5bb 272	. . . . 5 ⊢ (𝑦 = 𝐵 → ({𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵))
14	13	rexsng 4190	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵))
15		breq1 4616	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝐵 ↔ 𝐵𝑅𝐵))
16	15	notbid 308	. . . . 5 ⊢ (𝑥 = 𝐵 → (¬ 𝑥𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
17	16	ralsng 4189	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ {𝐵} ¬ 𝑥𝑅𝐵 ↔ ¬ 𝐵𝑅𝐵))
18	14, 17	bitrd 268	. . 3 ⊢ (𝐵 ∈ 𝐴 → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵))
19	18	adantl 482	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (∃𝑦 ∈ {𝐵} {𝑥 ∈ {𝐵} ∣ 𝑥𝑅𝑦} = ∅ ↔ ¬ 𝐵𝑅𝐵))
20	8, 19	mpbid 222	1 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  ∃wrex 2908  {crab 2911   ⊆ wss 3555  ∅c0 3891  {csn 4148   class class class wbr 4613   Fr wfr 5030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-fr 5033

This theorem is referenced by:  efrirr  5055  predfrirr  5668  dfwe2  6928  bnj1417  30814  efrunt  31295  ifr0  38133