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Mirrors > Home > NFE Home > Th. List > ncseqnc | GIF version |
Description: A cardinal is equal to the cardinality of a set iff it contains the set. (Contributed by SF, 24-Feb-2015.) |
Ref | Expression |
---|---|
ncseqnc | ⊢ (A ∈ NC → (A = Nc X ↔ X ∈ A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elncs 6119 | . 2 ⊢ (A ∈ NC ↔ ∃y A = Nc y) | |
2 | vex 2862 | . . . . . . . . . . 11 ⊢ y ∈ V | |
3 | 2 | ncid 6123 | . . . . . . . . . 10 ⊢ y ∈ Nc y |
4 | eleq2 2414 | . . . . . . . . . 10 ⊢ ( Nc X = Nc y → (y ∈ Nc X ↔ y ∈ Nc y)) | |
5 | 3, 4 | mpbiri 224 | . . . . . . . . 9 ⊢ ( Nc X = Nc y → y ∈ Nc X) |
6 | df-nc 6101 | . . . . . . . . 9 ⊢ Nc X = [X] ≈ | |
7 | 5, 6 | syl6eleq 2443 | . . . . . . . 8 ⊢ ( Nc X = Nc y → y ∈ [X] ≈ ) |
8 | ecexr 5950 | . . . . . . . 8 ⊢ (y ∈ [X] ≈ → X ∈ V) | |
9 | 7, 8 | syl 15 | . . . . . . 7 ⊢ ( Nc X = Nc y → X ∈ V) |
10 | brex 4689 | . . . . . . . 8 ⊢ (X ≈ y → (X ∈ V ∧ y ∈ V)) | |
11 | 10 | simpld 445 | . . . . . . 7 ⊢ (X ≈ y → X ∈ V) |
12 | ener 6039 | . . . . . . . . . 10 ⊢ ≈ Er V | |
13 | 12 | a1i 10 | . . . . . . . . 9 ⊢ (X ∈ V → ≈ Er V) |
14 | dmen 6041 | . . . . . . . . . 10 ⊢ dom ≈ = V | |
15 | 14 | a1i 10 | . . . . . . . . 9 ⊢ (X ∈ V → dom ≈ = V) |
16 | id 19 | . . . . . . . . 9 ⊢ (X ∈ V → X ∈ V) | |
17 | 2 | a1i 10 | . . . . . . . . 9 ⊢ (X ∈ V → y ∈ V) |
18 | 13, 15, 16, 17 | erth 5968 | . . . . . . . 8 ⊢ (X ∈ V → (X ≈ y ↔ [X] ≈ = [y] ≈ )) |
19 | df-nc 6101 | . . . . . . . . 9 ⊢ Nc y = [y] ≈ | |
20 | 6, 19 | eqeq12i 2366 | . . . . . . . 8 ⊢ ( Nc X = Nc y ↔ [X] ≈ = [y] ≈ ) |
21 | 18, 20 | syl6rbbr 255 | . . . . . . 7 ⊢ (X ∈ V → ( Nc X = Nc y ↔ X ≈ y)) |
22 | 9, 11, 21 | pm5.21nii 342 | . . . . . 6 ⊢ ( Nc X = Nc y ↔ X ≈ y) |
23 | eqcom 2355 | . . . . . 6 ⊢ ( Nc y = Nc X ↔ Nc X = Nc y) | |
24 | elnc 6125 | . . . . . 6 ⊢ (X ∈ Nc y ↔ X ≈ y) | |
25 | 22, 23, 24 | 3bitr4i 268 | . . . . 5 ⊢ ( Nc y = Nc X ↔ X ∈ Nc y) |
26 | 25 | a1i 10 | . . . 4 ⊢ (A = Nc y → ( Nc y = Nc X ↔ X ∈ Nc y)) |
27 | eqeq1 2359 | . . . 4 ⊢ (A = Nc y → (A = Nc X ↔ Nc y = Nc X)) | |
28 | eleq2 2414 | . . . 4 ⊢ (A = Nc y → (X ∈ A ↔ X ∈ Nc y)) | |
29 | 26, 27, 28 | 3bitr4d 276 | . . 3 ⊢ (A = Nc y → (A = Nc X ↔ X ∈ A)) |
30 | 29 | exlimiv 1634 | . 2 ⊢ (∃y A = Nc y → (A = Nc X ↔ X ∈ A)) |
31 | 1, 30 | sylbi 187 | 1 ⊢ (A ∈ NC → (A = Nc X ↔ X ∈ A)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∃wex 1541 = wceq 1642 ∈ wcel 1710 Vcvv 2859 class class class wbr 4639 dom cdm 4772 Er cer 5898 [cec 5945 ≈ cen 6028 NC cncs 6088 Nc cnc 6091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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-1st 4723 df-swap 4724 df-sset 4725 df-co 4726 df-ima 4727 df-si 4728 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 df-2nd 4797 df-txp 5736 df-ins2 5750 df-ins3 5752 df-image 5754 df-ins4 5756 df-si3 5758 df-funs 5760 df-fns 5762 df-trans 5899 df-sym 5908 df-er 5909 df-ec 5947 df-qs 5951 df-en 6029 df-ncs 6098 df-nc 6101 |
This theorem is referenced by: eqnc2 6129 df1c3 6140 peano4nc 6150 ncspw1eu 6159 ce0addcnnul 6179 ce0nnulb 6182 ceclb 6183 ce0ncpw1 6185 ce0 6190 ce2 6192 sbth 6206 dflec2 6210 taddc 6229 letc 6231 ce0lenc1 6239 finnc 6243 tcfnex 6244 ncfin 6247 addcdi 6250 |
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