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Theorem sbth 6206
 Description: The Schroder-Bernstein Theorem. This theorem gives the antisymmetry law for cardinal less than or equal. Translated out, it means that, if A is no larger than B and B is no larger than A, then Nc A = Nc B. Theorem XI.2.20 of [Rosser] p. 376. (Contributed by SF, 11-Mar-2015.)
Assertion
Ref Expression
sbth ((A NC B NC ) → ((Ac B Bc A) → A = B))

Proof of Theorem sbth
Dummy variables a b d g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brlecg 6112 . . . 4 ((A NC B NC ) → (Ac Bg A b B g b))
2 brlecg 6112 . . . . 5 ((B NC A NC ) → (Bc Ad B a A d a))
32ancoms 439 . . . 4 ((A NC B NC ) → (Bc Ad B a A d a))
41, 3anbi12d 691 . . 3 ((A NC B NC ) → ((Ac B Bc A) ↔ (g A b B g b d B a A d a)))
5 reeanv 2778 . . . . 5 (b B a A (g b d a) ↔ (b B g b a A d a))
652rexbii 2641 . . . 4 (g A d B b B a A (g b d a) ↔ g A d B (b B g b a A d a))
7 reeanv 2778 . . . 4 (g A d B (b B g b a A d a) ↔ (g A b B g b d B a A d a))
86, 7bitri 240 . . 3 (g A d B b B a A (g b d a) ↔ (g A b B g b d B a A d a))
94, 8syl6bbr 254 . 2 ((A NC B NC ) → ((Ac B Bc A) ↔ g A d B b B a A (g b d a)))
10 ncseqnc 6128 . . . . . 6 (A NC → (A = Nc gg A))
11 ncseqnc 6128 . . . . . 6 (B NC → (B = Nc dd B))
1210, 11bi2anan9 843 . . . . 5 ((A NC B NC ) → ((A = Nc g B = Nc d) ↔ (g A d B)))
1312biimpar 471 . . . 4 (((A NC B NC ) (g A d B)) → (A = Nc g B = Nc d))
14 simplr 731 . . . . . . . . . . 11 (((bd ag) (g b d a)) → ag)
15 ensym 6037 . . . . . . . . . . 11 (agga)
1614, 15sylib 188 . . . . . . . . . 10 (((bd ag) (g b d a)) → ga)
17 simprl 732 . . . . . . . . . . 11 (((bd ag) (g b d a)) → g b)
18 simpll 730 . . . . . . . . . . 11 (((bd ag) (g b d a)) → bd)
19 simprr 733 . . . . . . . . . . 11 (((bd ag) (g b d a)) → d a)
20 sbthlem3 6205 . . . . . . . . . . 11 (((ag g b) (bd d a)) → ab)
2114, 17, 18, 19, 20syl22anc 1183 . . . . . . . . . 10 (((bd ag) (g b d a)) → ab)
22 entr 6038 . . . . . . . . . 10 ((ga ab) → gb)
2316, 21, 22syl2anc 642 . . . . . . . . 9 (((bd ag) (g b d a)) → gb)
24 entr 6038 . . . . . . . . 9 ((gb bd) → gd)
2523, 18, 24syl2anc 642 . . . . . . . 8 (((bd ag) (g b d a)) → gd)
2625ex 423 . . . . . . 7 ((bd ag) → ((g b d a) → gd))
27 elnc 6125 . . . . . . . 8 (b Nc dbd)
28 elnc 6125 . . . . . . . 8 (a Nc gag)
2927, 28anbi12i 678 . . . . . . 7 ((b Nc d a Nc g) ↔ (bd ag))
30 vex 2862 . . . . . . . . 9 g V
3130eqnc 6127 . . . . . . . 8 ( Nc g = Nc dgd)
3231imbi2i 303 . . . . . . 7 (((g b d a) → Nc g = Nc d) ↔ ((g b d a) → gd))
3326, 29, 323imtr4i 257 . . . . . 6 ((b Nc d a Nc g) → ((g b d a) → Nc g = Nc d))
3433rexlimivv 2743 . . . . 5 (b Nc da Nc g(g b d a) → Nc g = Nc d)
35 rexeq 2808 . . . . . . 7 (B = Nc d → (b B a A (g b d a) ↔ b Nc da A (g b d a)))
36 rexeq 2808 . . . . . . . 8 (A = Nc g → (a A (g b d a) ↔ a Nc g(g b d a)))
3736rexbidv 2635 . . . . . . 7 (A = Nc g → (b Nc da A (g b d a) ↔ b Nc da Nc g(g b d a)))
3835, 37sylan9bbr 681 . . . . . 6 ((A = Nc g B = Nc d) → (b B a A (g b d a) ↔ b Nc da Nc g(g b d a)))
39 eqeq12 2365 . . . . . 6 ((A = Nc g B = Nc d) → (A = BNc g = Nc d))
4038, 39imbi12d 311 . . . . 5 ((A = Nc g B = Nc d) → ((b B a A (g b d a) → A = B) ↔ (b Nc da Nc g(g b d a) → Nc g = Nc d)))
4134, 40mpbiri 224 . . . 4 ((A = Nc g B = Nc d) → (b B a A (g b d a) → A = B))
4213, 41syl 15 . . 3 (((A NC B NC ) (g A d B)) → (b B a A (g b d a) → A = B))
4342rexlimdvva 2745 . 2 ((A NC B NC ) → (g A d B b B a A (g b d a) → A = B))
449, 43sylbid 206 1 ((A NC B NC ) → ((Ac B Bc A) → A = B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ⊆ wss 3257   class class class wbr 4639   ≈ 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-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-nc 6101 This theorem is referenced by:  ltlenlec  6207  leltctr  6212  lecponc  6213  nclenn  6249  ncvsq  6256
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