Theorem frind 9751

Description: A subclass of a well-founded class 𝐴 with the property that whenever it contains all predecessors of an element of 𝐴 it also contains that element, is equal to 𝐴. Compare wfi 6351 and tfi 7846, which are special cases of this theorem that do not require the axiom of infinity. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Assertion

Ref	Expression
frind	⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 = 𝐵)

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝑅

Proof of Theorem frind

Step	Hyp	Ref	Expression
1		ssdif0 4363	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
2	1	necon3bbii 2987	. . . . . 6 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) ≠ ∅)
3		difss 4131	. . . . . . 7 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
4		frmin 9750	. . . . . . . . 9 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ ((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∖ 𝐵) ≠ ∅)) → ∃𝑦 ∈ (𝐴 ∖ 𝐵)Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅)
5		eldif 3958	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
6	5	anbi1i 623	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (𝐴 ∖ 𝐵) ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵) ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅))
7		anass 468	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵) ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ (𝑦 ∈ 𝐴 ∧ (¬ 𝑦 ∈ 𝐵 ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅)))
8		ancom 460	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑦 ∈ 𝐵 ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ (Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ∧ ¬ 𝑦 ∈ 𝐵))
9		indif2 4270	. . . . . . . . . . . . . . . . . 18 ⊢ ((◡𝑅 “ {𝑦}) ∩ (𝐴 ∖ 𝐵)) = (((◡𝑅 “ {𝑦}) ∩ 𝐴) ∖ 𝐵)
10		df-pred 6300	. . . . . . . . . . . . . . . . . . 19 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ((𝐴 ∖ 𝐵) ∩ (◡𝑅 “ {𝑦}))
11		incom 4201	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∖ 𝐵) ∩ (◡𝑅 “ {𝑦})) = ((◡𝑅 “ {𝑦}) ∩ (𝐴 ∖ 𝐵))
12	10, 11	eqtri 2759	. . . . . . . . . . . . . . . . . 18 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ((◡𝑅 “ {𝑦}) ∩ (𝐴 ∖ 𝐵))
13		df-pred 6300	. . . . . . . . . . . . . . . . . . . 20 ⊢ Pred(𝑅, 𝐴, 𝑦) = (𝐴 ∩ (◡𝑅 “ {𝑦}))
14		incom 4201	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑦})) = ((◡𝑅 “ {𝑦}) ∩ 𝐴)
15	13, 14	eqtri 2759	. . . . . . . . . . . . . . . . . . 19 ⊢ Pred(𝑅, 𝐴, 𝑦) = ((◡𝑅 “ {𝑦}) ∩ 𝐴)
16	15	difeq1i 4118	. . . . . . . . . . . . . . . . . 18 ⊢ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = (((◡𝑅 “ {𝑦}) ∩ 𝐴) ∖ 𝐵)
17	9, 12, 16	3eqtr4i 2769	. . . . . . . . . . . . . . . . 17 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵)
18	17	eqeq1i 2736	. . . . . . . . . . . . . . . 16 ⊢ (Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ↔ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = ∅)
19		ssdif0 4363	. . . . . . . . . . . . . . . 16 ⊢ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ↔ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = ∅)
20	18, 19	bitr4i 278	. . . . . . . . . . . . . . 15 ⊢ (Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵)
21	20	anbi1i 623	. . . . . . . . . . . . . 14 ⊢ ((Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ∧ ¬ 𝑦 ∈ 𝐵) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵))
22	8, 21	bitri 275	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑦 ∈ 𝐵 ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵))
23	22	anbi2i 622	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝑦 ∈ 𝐵 ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅)) ↔ (𝑦 ∈ 𝐴 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵)))
24	6, 7, 23	3bitri 297	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (𝐴 ∖ 𝐵) ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ (𝑦 ∈ 𝐴 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵)))
25	24	rexbii2 3089	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ (𝐴 ∖ 𝐵)Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵))
26		rexanali 3101	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵) ↔ ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))
27	25, 26	bitri 275	. . . . . . . . 9 ⊢ (∃𝑦 ∈ (𝐴 ∖ 𝐵)Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ↔ ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))
28	4, 27	sylib 217	. . . . . . . 8 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ ((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∖ 𝐵) ≠ ∅)) → ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))
29	28	ex 412	. . . . . . 7 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∖ 𝐵) ≠ ∅) → ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵)))
30	3, 29	mpani 693	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → ((𝐴 ∖ 𝐵) ≠ ∅ → ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵)))
31	2, 30	biimtrid 241	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (¬ 𝐴 ⊆ 𝐵 → ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵)))
32	31	con4d 115	. . . 4 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵) → 𝐴 ⊆ 𝐵))
33	32	imp 406	. . 3 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵)) → 𝐴 ⊆ 𝐵)
34	33	adantrl 713	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 ⊆ 𝐵)
35		simprl 768	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐵 ⊆ 𝐴)
36	34, 35	eqssd 3999	1 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  ∀wral 3060  ∃wrex 3069   ∖ cdif 3945   ∩ cin 3947   ⊆ wss 3948  ∅c0 4322  {csn 4628   Fr wfr 5628   Se wse 5629  ^◡ccnv 5675   “ cima 5679  Predcpred 6299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729  ax-inf2 9642

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-oadd 8476  df-ttrcl 9709

This theorem is referenced by:  frinsg  9752