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Theorem dffin7-2 9220

Description: Class form of isfin7-2 9218. (Contributed by Mario Carneiro, 17-May-2015.)

**Assertion**
Ref	Expression
dffin7-2	⊢ Fin^VII = (Fin ∪ (V ∖ dom card))

**Proof of Theorem dffin7-2**
Step	Hyp	Ref	Expression
1		imor 428	. . 3 ⊢ ((𝑥 ∈ dom card → 𝑥 ∈ Fin) ↔ (¬ 𝑥 ∈ dom card ∨ 𝑥 ∈ Fin))
2		vex 3203	. . . 4 ⊢ 𝑥 ∈ V
3		isfin7-2 9218	. . . 4 ⊢ (𝑥 ∈ V → (𝑥 ∈ Fin^VII ↔ (𝑥 ∈ dom card → 𝑥 ∈ Fin)))
4	2, 3	ax-mp 5	. . 3 ⊢ (𝑥 ∈ Fin^VII ↔ (𝑥 ∈ dom card → 𝑥 ∈ Fin))
5		elun 3753	. . . 4 ⊢ (𝑥 ∈ (Fin ∪ (V ∖ dom card)) ↔ (𝑥 ∈ Fin ∨ 𝑥 ∈ (V ∖ dom card)))
6		orcom 402	. . . 4 ⊢ ((𝑥 ∈ Fin ∨ 𝑥 ∈ (V ∖ dom card)) ↔ (𝑥 ∈ (V ∖ dom card) ∨ 𝑥 ∈ Fin))
7		eldif 3584	. . . . . 6 ⊢ (𝑥 ∈ (V ∖ dom card) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ dom card))
8	2, 7	mpbiran 953	. . . . 5 ⊢ (𝑥 ∈ (V ∖ dom card) ↔ ¬ 𝑥 ∈ dom card)
9	8	orbi1i 542	. . . 4 ⊢ ((𝑥 ∈ (V ∖ dom card) ∨ 𝑥 ∈ Fin) ↔ (¬ 𝑥 ∈ dom card ∨ 𝑥 ∈ Fin))
10	5, 6, 9	3bitri 286	. . 3 ⊢ (𝑥 ∈ (Fin ∪ (V ∖ dom card)) ↔ (¬ 𝑥 ∈ dom card ∨ 𝑥 ∈ Fin))
11	1, 4, 10	3bitr4i 292	. 2 ⊢ (𝑥 ∈ Fin^VII ↔ 𝑥 ∈ (Fin ∪ (V ∖ dom card)))
12	11	eqriv 2619	1 ⊢ Fin^VII = (Fin ∪ (V ∖ dom card))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 dom cdm 5114 Fincfn 7955 cardccrd 8761 Fin^VIIcfin7 9106

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-fin7 9113

This theorem is referenced by: dfacfin7 9221

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