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Mirrors > Home > NFE Home > Th. List > lecncvg | GIF version |
Description: The cardinality of V is a maximal element of cardinal less than or equal. Theorem XI.2.16 of [Rosser] p. 376. (Contributed by SF, 4-Mar-2015.) |
Ref | Expression |
---|---|
lecncvg | ⊢ ((A ∈ V ∧ A ≠ ∅) → A ≤c Nc V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vvex 4109 | . . . . . . . 8 ⊢ V ∈ V | |
2 | 1 | ncid 6123 | . . . . . . 7 ⊢ V ∈ Nc V |
3 | ssv 3291 | . . . . . . 7 ⊢ x ⊆ V | |
4 | sseq2 3293 | . . . . . . . 8 ⊢ (y = V → (x ⊆ y ↔ x ⊆ V)) | |
5 | 4 | rspcev 2955 | . . . . . . 7 ⊢ ((V ∈ Nc V ∧ x ⊆ V) → ∃y ∈ Nc Vx ⊆ y) |
6 | 2, 3, 5 | mp2an 653 | . . . . . 6 ⊢ ∃y ∈ Nc Vx ⊆ y |
7 | 6 | jctr 526 | . . . . 5 ⊢ (x ∈ A → (x ∈ A ∧ ∃y ∈ Nc Vx ⊆ y)) |
8 | 7 | eximi 1576 | . . . 4 ⊢ (∃x x ∈ A → ∃x(x ∈ A ∧ ∃y ∈ Nc Vx ⊆ y)) |
9 | n0 3559 | . . . 4 ⊢ (A ≠ ∅ ↔ ∃x x ∈ A) | |
10 | df-rex 2620 | . . . 4 ⊢ (∃x ∈ A ∃y ∈ Nc Vx ⊆ y ↔ ∃x(x ∈ A ∧ ∃y ∈ Nc Vx ⊆ y)) | |
11 | 8, 9, 10 | 3imtr4i 257 | . . 3 ⊢ (A ≠ ∅ → ∃x ∈ A ∃y ∈ Nc Vx ⊆ y) |
12 | 11 | adantl 452 | . 2 ⊢ ((A ∈ V ∧ A ≠ ∅) → ∃x ∈ A ∃y ∈ Nc Vx ⊆ y) |
13 | ncex 6117 | . . . 4 ⊢ Nc V ∈ V | |
14 | brlecg 6112 | . . . 4 ⊢ ((A ∈ V ∧ Nc V ∈ V) → (A ≤c Nc V ↔ ∃x ∈ A ∃y ∈ Nc Vx ⊆ y)) | |
15 | 13, 14 | mpan2 652 | . . 3 ⊢ (A ∈ V → (A ≤c Nc V ↔ ∃x ∈ A ∃y ∈ Nc Vx ⊆ y)) |
16 | 15 | adantr 451 | . 2 ⊢ ((A ∈ V ∧ A ≠ ∅) → (A ≤c Nc V ↔ ∃x ∈ A ∃y ∈ Nc Vx ⊆ y)) |
17 | 12, 16 | mpbird 223 | 1 ⊢ ((A ∈ V ∧ A ≠ ∅) → A ≤c Nc V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∃wex 1541 ∈ wcel 1710 ≠ wne 2516 ∃wrex 2615 Vcvv 2859 ⊆ wss 3257 ∅c0 3550 class class class wbr 4639 ≤c clec 6089 Nc cnc 6091 |
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-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-ec 5947 df-en 6029 df-lec 6099 df-nc 6101 |
This theorem is referenced by: ncvsq 6256 |
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