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Theorem lenc 6223
 Description: Less than or equal condition for the cardinality of a number. (Contributed by SF, 18-Mar-2015.)
Hypothesis
Ref Expression
lenc.1 A V
Assertion
Ref Expression
lenc (M NC → (Mc Nc Ax M x A))
Distinct variable groups:   x,M   x,A

Proof of Theorem lenc
Dummy variables f g p q y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elncs 6119 . 2 (M NCy M = Nc y)
2 ncex 6117 . . . . . . 7 Nc y V
3 ncex 6117 . . . . . . 7 Nc A V
42, 3brlec 6113 . . . . . 6 ( Nc yc Nc Ap Nc yq Nc Ap q)
5 elnc 6125 . . . . . . . . . . 11 (p Nc ypy)
6 bren 6030 . . . . . . . . . . 11 (pyf f:p1-1-ontoy)
75, 6bitri 240 . . . . . . . . . 10 (p Nc yf f:p1-1-ontoy)
8 elnc 6125 . . . . . . . . . . 11 (q Nc AqA)
9 bren 6030 . . . . . . . . . . 11 (qAg g:q1-1-ontoA)
108, 9bitri 240 . . . . . . . . . 10 (q Nc Ag g:q1-1-ontoA)
117, 10anbi12i 678 . . . . . . . . 9 ((p Nc y q Nc A) ↔ (f f:p1-1-ontoy g g:q1-1-ontoA))
12 eeanv 1913 . . . . . . . . 9 (fg(f:p1-1-ontoy g:q1-1-ontoA) ↔ (f f:p1-1-ontoy g g:q1-1-ontoA))
1311, 12bitr4i 243 . . . . . . . 8 ((p Nc y q Nc A) ↔ fg(f:p1-1-ontoy g:q1-1-ontoA))
14 f1of1 5286 . . . . . . . . . . . . . . . 16 (g:q1-1-ontoAg:q1-1A)
15143ad2ant2 977 . . . . . . . . . . . . . . 15 ((f:p1-1-ontoy g:q1-1-ontoA p q) → g:q1-1A)
16 simp3 957 . . . . . . . . . . . . . . 15 ((f:p1-1-ontoy g:q1-1-ontoA p q) → p q)
17 f1ores 5300 . . . . . . . . . . . . . . 15 ((g:q1-1A p q) → (g p):p1-1-onto→(gp))
1815, 16, 17syl2anc 642 . . . . . . . . . . . . . 14 ((f:p1-1-ontoy g:q1-1-ontoA p q) → (g p):p1-1-onto→(gp))
19 f1ocnv 5299 . . . . . . . . . . . . . . 15 (f:p1-1-ontoyf:y1-1-ontop)
20193ad2ant1 976 . . . . . . . . . . . . . 14 ((f:p1-1-ontoy g:q1-1-ontoA p q) → f:y1-1-ontop)
21 f1oco 5308 . . . . . . . . . . . . . 14 (((g p):p1-1-onto→(gp) f:y1-1-ontop) → ((g p) f):y1-1-onto→(gp))
2218, 20, 21syl2anc 642 . . . . . . . . . . . . 13 ((f:p1-1-ontoy g:q1-1-ontoA p q) → ((g p) f):y1-1-onto→(gp))
23 f1ocnv 5299 . . . . . . . . . . . . 13 (((g p) f):y1-1-onto→(gp) → ((g p) f):(gp)–1-1-ontoy)
24 vex 2862 . . . . . . . . . . . . . . . . 17 g V
25 vex 2862 . . . . . . . . . . . . . . . . 17 p V
2624, 25resex 5117 . . . . . . . . . . . . . . . 16 (g p) V
27 vex 2862 . . . . . . . . . . . . . . . . 17 f V
2827cnvex 5102 . . . . . . . . . . . . . . . 16 f V
2926, 28coex 4750 . . . . . . . . . . . . . . 15 ((g p) f) V
3029cnvex 5102 . . . . . . . . . . . . . 14 ((g p) f) V
3130f1oen 6033 . . . . . . . . . . . . 13 (((g p) f):(gp)–1-1-ontoy → (gp) ≈ y)
3222, 23, 313syl 18 . . . . . . . . . . . 12 ((f:p1-1-ontoy g:q1-1-ontoA p q) → (gp) ≈ y)
33 elnc 6125 . . . . . . . . . . . 12 ((gp) Nc y ↔ (gp) ≈ y)
3432, 33sylibr 203 . . . . . . . . . . 11 ((f:p1-1-ontoy g:q1-1-ontoA p q) → (gp) Nc y)
35 imass2 5024 . . . . . . . . . . . . 13 (p q → (gp) (gq))
36353ad2ant3 978 . . . . . . . . . . . 12 ((f:p1-1-ontoy g:q1-1-ontoA p q) → (gp) (gq))
37 f1ofo 5293 . . . . . . . . . . . . . 14 (g:q1-1-ontoAg:qontoA)
38 foima 5274 . . . . . . . . . . . . . 14 (g:qontoA → (gq) = A)
3937, 38syl 15 . . . . . . . . . . . . 13 (g:q1-1-ontoA → (gq) = A)
40393ad2ant2 977 . . . . . . . . . . . 12 ((f:p1-1-ontoy g:q1-1-ontoA p q) → (gq) = A)
4136, 40sseqtrd 3307 . . . . . . . . . . 11 ((f:p1-1-ontoy g:q1-1-ontoA p q) → (gp) A)
42 sseq1 3292 . . . . . . . . . . . 12 (x = (gp) → (x A ↔ (gp) A))
4342rspcev 2955 . . . . . . . . . . 11 (((gp) Nc y (gp) A) → x Nc yx A)
4434, 41, 43syl2anc 642 . . . . . . . . . 10 ((f:p1-1-ontoy g:q1-1-ontoA p q) → x Nc yx A)
45443expia 1153 . . . . . . . . 9 ((f:p1-1-ontoy g:q1-1-ontoA) → (p qx Nc yx A))
4645exlimivv 1635 . . . . . . . 8 (fg(f:p1-1-ontoy g:q1-1-ontoA) → (p qx Nc yx A))
4713, 46sylbi 187 . . . . . . 7 ((p Nc y q Nc A) → (p qx Nc yx A))
4847rexlimivv 2743 . . . . . 6 (p Nc yq Nc Ap qx Nc yx A)
494, 48sylbi 187 . . . . 5 ( Nc yc Nc Ax Nc yx A)
50 vex 2862 . . . . . . . 8 x V
51 lenc.1 . . . . . . . 8 A V
5250, 51nclec 6195 . . . . . . 7 (x ANc xc Nc A)
5350eqnc 6127 . . . . . . . . 9 ( Nc x = Nc yxy)
54 elnc 6125 . . . . . . . . 9 (x Nc yxy)
5553, 54bitr4i 243 . . . . . . . 8 ( Nc x = Nc yx Nc y)
56 breq1 4642 . . . . . . . 8 ( Nc x = Nc y → ( Nc xc Nc ANc yc Nc A))
5755, 56sylbir 204 . . . . . . 7 (x Nc y → ( Nc xc Nc ANc yc Nc A))
5852, 57syl5ib 210 . . . . . 6 (x Nc y → (x ANc yc Nc A))
5958rexlimiv 2732 . . . . 5 (x Nc yx ANc yc Nc A)
6049, 59impbii 180 . . . 4 ( Nc yc Nc Ax Nc yx A)
61 breq1 4642 . . . . 5 (M = Nc y → (Mc Nc ANc yc Nc A))
62 rexeq 2808 . . . . 5 (M = Nc y → (x M x Ax Nc yx A))
6361, 62bibi12d 312 . . . 4 (M = Nc y → ((Mc Nc Ax M x A) ↔ ( Nc yc Nc Ax Nc yx A)))
6460, 63mpbiri 224 . . 3 (M = Nc y → (Mc Nc Ax M x A))
6564exlimiv 1634 . 2 (y M = Nc y → (Mc Nc Ax M x A))
661, 65sylbi 187 1 (M NC → (Mc Nc Ax M x A))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859   ⊆ wss 3257   class class class wbr 4639   ∘ ccom 4721   “ cima 4722  ◡ccnv 4771   ↾ cres 4774  –1-1→wf1 4778  –onto→wfo 4779  –1-1-onto→wf1o 4780   ≈ cen 6028   NC cncs 6088   ≤c clec 6089   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-lec 6099  df-nc 6101 This theorem is referenced by:  ce0lenc1  6239  nchoicelem13  6301
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