Theorem nclec 6196

Description: A relationship between cardinality, subset, and cardinal less than. (Contributed by SF, 17-Mar-2015.)

Hypotheses

Ref	Expression
nclec.1	⊢ A ∈ V
nclec.2	⊢ B ∈ V

Assertion

Ref	Expression
nclec	⊢ (A ⊆ B → Nc A ≤c Nc B)

Proof of Theorem nclec

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nclec.1	. . . 4 ⊢ A ∈ V
2	1	ncid 6124	. . 3 ⊢ A ∈ Nc A
3		nclec.2	. . . 4 ⊢ B ∈ V
4	3	ncid 6124	. . 3 ⊢ B ∈ Nc B
5		sseq1 3293	. . . 4 ⊢ (x = A → (x ⊆ y ↔ A ⊆ y))
6		sseq2 3294	. . . 4 ⊢ (y = B → (A ⊆ y ↔ A ⊆ B))
7	5, 6	rspc2ev 2964	. . 3 ⊢ ((A ∈ Nc A ∧ B ∈ Nc B ∧ A ⊆ B) → ∃x ∈ Nc A∃y ∈ Nc Bx ⊆ y)
8	2, 4, 7	mp3an12 1267	. 2 ⊢ (A ⊆ B → ∃x ∈ Nc A∃y ∈ Nc Bx ⊆ y)
9		ncex 6118	. . 3 ⊢ Nc A ∈ V
10		ncex 6118	. . 3 ⊢ Nc B ∈ V
11	9, 10	brlec 6114	. 2 ⊢ ( Nc A ≤c Nc B ↔ ∃x ∈ Nc A∃y ∈ Nc Bx ⊆ y)
12	8, 11	sylibr 203	1 ⊢ (A ⊆ B → Nc A ≤c Nc B)

Syntax hints:   → wi 4   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860   ⊆ wss 3258   class class class wbr 4640   ≤_c clec 6090   Nc cnc 6092

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-2nd 4798  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-ec 5948  df-en 6030  df-lec 6100  df-nc 6102

This theorem is referenced by:  lenc  6224  ce0lenc1  6240  0lt1c  6259