Theorem leltctr 6212

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 6211	. . . . . 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 4642	. . . . . . . . . . 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 6206	. . . . . . . . . . . 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 2554	. . . . . 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 6114	. . 3 ⊢ (B <c C ↔ (B ≤c C ∧ B ≠ C))
19		brltc 6114	. . 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 2516   class class class wbr 4639   NC cncs 6088   ≤_c clec 6089   <_c cltc 6090

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-fix 5740  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-clos1 5873  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-ltc 6100  df-nc 6101

This theorem is referenced by: (None)