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Theorem ceexlem1 6173

Description: Lemma for ceex 6174. Set up part of the stratification. (Contributed by SF, 6-Mar-2015.)

**Assertion**
Ref	Expression
ceexlem1	⊢ (⟨{{a}}, n⟩ ∈ ( S ∘ SI Pw1Fn ) ↔ ℘₁a ∈ n)

Distinct variable group: n,a

Proof of Theorem ceexlem1 Dummy variables t u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 4111	. . . . . . . 8 ⊢ {a} ∈ V
2	1	brsnsi1 5775	. . . . . . 7 ⊢ ({{a}} SI Pw1Fn u ↔ ∃t(u = {t} ∧ {a} Pw1Fn t))
3	2	anbi1i 676	. . . . . 6 ⊢ (({{a}} SI Pw1Fn u ∧ u S n) ↔ (∃t(u = {t} ∧ {a} Pw1Fn t) ∧ u S n))
4		19.41v 1901	. . . . . 6 ⊢ (∃t((u = {t} ∧ {a} Pw1Fn t) ∧ u S n) ↔ (∃t(u = {t} ∧ {a} Pw1Fn t) ∧ u S n))
5		anass 630	. . . . . . 7 ⊢ (((u = {t} ∧ {a} Pw1Fn t) ∧ u S n) ↔ (u = {t} ∧ ({a} Pw1Fn t ∧ u S n)))
6	5	exbii 1582	. . . . . 6 ⊢ (∃t((u = {t} ∧ {a} Pw1Fn t) ∧ u S n) ↔ ∃t(u = {t} ∧ ({a} Pw1Fn t ∧ u S n)))
7	3, 4, 6	3bitr2i 264	. . . . 5 ⊢ (({{a}} SI Pw1Fn u ∧ u S n) ↔ ∃t(u = {t} ∧ ({a} Pw1Fn t ∧ u S n)))
8	7	exbii 1582	. . . 4 ⊢ (∃u({{a}} SI Pw1Fn u ∧ u S n) ↔ ∃u∃t(u = {t} ∧ ({a} Pw1Fn t ∧ u S n)))
9		excom 1741	. . . 4 ⊢ (∃u∃t(u = {t} ∧ ({a} Pw1Fn t ∧ u S n)) ↔ ∃t∃u(u = {t} ∧ ({a} Pw1Fn t ∧ u S n)))
10	8, 9	bitri 240	. . 3 ⊢ (∃u({{a}} SI Pw1Fn u ∧ u S n) ↔ ∃t∃u(u = {t} ∧ ({a} Pw1Fn t ∧ u S n)))
11		snex 4111	. . . . . 6 ⊢ {t} ∈ V
12		breq1 4642	. . . . . . 7 ⊢ (u = {t} → (u S n ↔ {t} S n))
13	12	anbi2d 684	. . . . . 6 ⊢ (u = {t} → (({a} Pw1Fn t ∧ u S n) ↔ ({a} Pw1Fn t ∧ {t} S n)))
14	11, 13	ceqsexv 2894	. . . . 5 ⊢ (∃u(u = {t} ∧ ({a} Pw1Fn t ∧ u S n)) ↔ ({a} Pw1Fn t ∧ {t} S n))
15		vex 2862	. . . . . . 7 ⊢ a ∈ V
16	15	brpw1fn 5854	. . . . . 6 ⊢ ({a} Pw1Fn t ↔ t = ℘₁a)
17		vex 2862	. . . . . . 7 ⊢ t ∈ V
18		vex 2862	. . . . . . 7 ⊢ n ∈ V
19	17, 18	brssetsn 4759	. . . . . 6 ⊢ ({t} S n ↔ t ∈ n)
20	16, 19	anbi12i 678	. . . . 5 ⊢ (({a} Pw1Fn t ∧ {t} S n) ↔ (t = ℘₁a ∧ t ∈ n))
21	14, 20	bitri 240	. . . 4 ⊢ (∃u(u = {t} ∧ ({a} Pw1Fn t ∧ u S n)) ↔ (t = ℘₁a ∧ t ∈ n))
22	21	exbii 1582	. . 3 ⊢ (∃t∃u(u = {t} ∧ ({a} Pw1Fn t ∧ u S n)) ↔ ∃t(t = ℘₁a ∧ t ∈ n))
23	10, 22	bitri 240	. 2 ⊢ (∃u({{a}} SI Pw1Fn u ∧ u S n) ↔ ∃t(t = ℘₁a ∧ t ∈ n))
24		opelco 4884	. 2 ⊢ (⟨{{a}}, n⟩ ∈ ( S ∘ SI Pw1Fn ) ↔ ∃u({{a}} SI Pw1Fn u ∧ u S n))
25		df-clel 2349	. 2 ⊢ (℘₁a ∈ n ↔ ∃t(t = ℘₁a ∧ t ∈ n))
26	23, 24, 25	3bitr4i 268	1 ⊢ (⟨{{a}}, n⟩ ∈ ( S ∘ SI Pw1Fn ) ↔ ℘₁a ∈ n)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 {csn 3737 ℘₁cpw1 4135 ⟨cop 4561 class class class wbr 4639 S csset 4719 SI csi 4720 ∘ ccom 4721 Pw1Fn cpw1fn 5765

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-sset 4725 df-co 4726 df-ima 4727 df-si 4728 df-id 4767 df-cnv 4785 df-rn 4786 df-dm 4787 df-fun 4789 df-fn 4790 df-fv 4795 df-mpt 5652 df-pw1fn 5766

This theorem is referenced by: ceex 6174

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