Theorem frind 9710

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 6338 and tfi 7835, 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 4321	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
2	1	necon3bbii 3006	. . . . . 6 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) ≠ ∅)
3		difss 4091	. . . . . . 7 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
4		frmin 9709	. . . . . . . . 9 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ ((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∖ 𝐵) ≠ ∅)) → ∃𝑦 ∈ (𝐴 ∖ 𝐵)Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅)
5		eldif 3916	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐴 ∖ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))
6	5	anbi1i 633	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (𝐴 ∖ 𝐵) ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵) ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅))
7		anass 472	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵) ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ (𝑦 ∈ 𝐴 ∧ (¬ 𝑦 ∈ 𝐵 ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅)))
8		ancom 464	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑦 ∈ 𝐵 ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ (Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ∧ ¬ 𝑦 ∈ 𝐵))
9		indif2 4235	. . . . . . . . . . . . . . . . . 18 ⊢ ((◡𝑅 “ {𝑦}) ∩ (𝐴 ∖ 𝐵)) = (((◡𝑅 “ {𝑦}) ∩ 𝐴) ∖ 𝐵)
10		df-pred 6290	. . . . . . . . . . . . . . . . . . 19 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ((𝐴 ∖ 𝐵) ∩ (◡𝑅 “ {𝑦}))
11		incom 4163	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∖ 𝐵) ∩ (◡𝑅 “ {𝑦})) = ((◡𝑅 “ {𝑦}) ∩ (𝐴 ∖ 𝐵))
12	10, 11	eqtri 2787	. . . . . . . . . . . . . . . . . 18 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ((◡𝑅 “ {𝑦}) ∩ (𝐴 ∖ 𝐵))
13		df-pred 6290	. . . . . . . . . . . . . . . . . . . 20 ⊢ Pred(𝑅, 𝐴, 𝑦) = (𝐴 ∩ (◡𝑅 “ {𝑦}))
14		incom 4163	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑦})) = ((◡𝑅 “ {𝑦}) ∩ 𝐴)
15	13, 14	eqtri 2787	. . . . . . . . . . . . . . . . . . 19 ⊢ Pred(𝑅, 𝐴, 𝑦) = ((◡𝑅 “ {𝑦}) ∩ 𝐴)
16	15	difeq1i 4078	. . . . . . . . . . . . . . . . . 18 ⊢ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = (((◡𝑅 “ {𝑦}) ∩ 𝐴) ∖ 𝐵)
17	9, 12, 16	3eqtr4i 2797	. . . . . . . . . . . . . . . . 17 ⊢ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵)
18	17	eqeq1i 2769	. . . . . . . . . . . . . . . 16 ⊢ (Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ↔ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = ∅)
19		ssdif0 4321	. . . . . . . . . . . . . . . 16 ⊢ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ↔ (Pred(𝑅, 𝐴, 𝑦) ∖ 𝐵) = ∅)
20	18, 19	bitr4i 280	. . . . . . . . . . . . . . 15 ⊢ (Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵)
21	20	anbi1i 633	. . . . . . . . . . . . . 14 ⊢ ((Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ∧ ¬ 𝑦 ∈ 𝐵) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵))
22	8, 21	bitri 277	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑦 ∈ 𝐵 ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵))
23	22	anbi2i 632	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ (¬ 𝑦 ∈ 𝐵 ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅)) ↔ (𝑦 ∈ 𝐴 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵)))
24	6, 7, 23	3bitri 299	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (𝐴 ∖ 𝐵) ∧ Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅) ↔ (𝑦 ∈ 𝐴 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵)))
25	24	rexbii2 3107	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ (𝐴 ∖ 𝐵)Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵))
26		rexanali 3118	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ ¬ 𝑦 ∈ 𝐵) ↔ ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))
27	25, 26	bitri 277	. . . . . . . . 9 ⊢ (∃𝑦 ∈ (𝐴 ∖ 𝐵)Pred(𝑅, (𝐴 ∖ 𝐵), 𝑦) = ∅ ↔ ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))
28	4, 27	sylib 220	. . . . . . . 8 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ ((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∖ 𝐵) ≠ ∅)) → ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))
29	28	ex 416	. . . . . . 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 706	. . . . . 6 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → ((𝐴 ∖ 𝐵) ≠ ∅ → ¬ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵)))
31	2, 30	biimtrid 244	. . . . 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 410	. . 3 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵)) → 𝐴 ⊆ 𝐵)
34	33	adantrl 726	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 ⊆ 𝐵)
35		simprl 780	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐵 ⊆ 𝐴)
36	34, 35	eqssd 3955	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 399   = wceq 1562   ∈ wcel 2144   ≠ wne 2959  ∀wral 3078  ∃wrex 3088   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  {csn 4584   Fr wfr 5599   Se wse 5600  ^◡ccnv 5648   “ cima 5652  Predcpred 6289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720  ax-inf2 9598

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-oadd 8443  df-ttrcl 9665

This theorem is referenced by:  frinsg  9711