Theorem leltctr 6213

Description: Transitivity law for cardinal less than or equal and less than. (Contributed by SF, 16-Mar-2015.)

Assertion

Ref	Expression
leltctr	⊢ ((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) → ((A ≤c B ∧ B <c C) → A <c C))

Proof of Theorem leltctr

Step	Hyp	Ref	Expression
1		lectr 6212	. . . . . 6 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) → ((A ≤c B ∧ B ≤c C) → A ≤c C))
2	1	expdimp 426	. . . . 5 ⊢ (((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) ∧ A ≤c B) → (B ≤c C → A ≤c C))
3	2	adantrd 454	. . . 4 ⊢ (((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) ∧ A ≤c B) → ((B ≤c C ∧ B ≠ C) → A ≤c C))
4		breq1 4643	. . . . . . . . . . 11 ⊢ (A = C → (A ≤c B ↔ C ≤c B))
5	4	anbi1d 685	. . . . . . . . . 10 ⊢ (A = C → ((A ≤c B ∧ B ≤c C) ↔ (C ≤c B ∧ B ≤c C)))
6	5	biimpac 472	. . . . . . . . 9 ⊢ (((A ≤c B ∧ B ≤c C) ∧ A = C) → (C ≤c B ∧ B ≤c C))
7		sbth 6207	. . . . . . . . . . . 12 ⊢ ((C ∈ NC ∧ B ∈ NC ) → ((C ≤c B ∧ B ≤c C) → C = B))
8	7	ancoms 439	. . . . . . . . . . 11 ⊢ ((B ∈ NC ∧ C ∈ NC ) → ((C ≤c B ∧ B ≤c C) → C = B))
9	8	3adant1 973	. . . . . . . . . 10 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) → ((C ≤c B ∧ B ≤c C) → C = B))
10		eqcom 2355	. . . . . . . . . 10 ⊢ (B = C ↔ C = B)
11	9, 10	syl6ibr 218	. . . . . . . . 9 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) → ((C ≤c B ∧ B ≤c C) → B = C))
12	6, 11	syl5 28	. . . . . . . 8 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) → (((A ≤c B ∧ B ≤c C) ∧ A = C) → B = C))
13	12	expdimp 426	. . . . . . 7 ⊢ (((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) ∧ (A ≤c B ∧ B ≤c C)) → (A = C → B = C))
14	13	necon3d 2555	. . . . . 6 ⊢ (((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) ∧ (A ≤c B ∧ B ≤c C)) → (B ≠ C → A ≠ C))
15	14	expr 598	. . . . 5 ⊢ (((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) ∧ A ≤c B) → (B ≤c C → (B ≠ C → A ≠ C)))
16	15	imp3a 420	. . . 4 ⊢ (((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) ∧ A ≤c B) → ((B ≤c C ∧ B ≠ C) → A ≠ C))
17	3, 16	jcad 519	. . 3 ⊢ (((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) ∧ A ≤c B) → ((B ≤c C ∧ B ≠ C) → (A ≤c C ∧ A ≠ C)))
18		brltc 6115	. . 3 ⊢ (B <c C ↔ (B ≤c C ∧ B ≠ C))
19		brltc 6115	. . 3 ⊢ (A <c C ↔ (A ≤c C ∧ A ≠ C))
20	17, 18, 19	3imtr4g 261	. 2 ⊢ (((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) ∧ A ≤c B) → (B <c C → A <c C))
21	20	expimpd 586	1 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) → ((A ≤c B ∧ B <c C) → A <c C))

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710   ≠ wne 2517   class class class wbr 4640   NC cncs 6089   ≤_c clec 6090   <_c cltc 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 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-fix 5741  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-clos1 5874  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-lec 6100  df-ltc 6101  df-nc 6102

This theorem is referenced by: (None)