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 ) → ((A ≤c B ∧ B ≤c A) → A = B))

Proof of Theorem sbth

Dummy variables a b d g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brlecg 6112	. . . 4 ⊢ ((A ∈ NC ∧ B ∈ NC ) → (A ≤c B ↔ ∃g ∈ A ∃b ∈ B g ⊆ b))
2		brlecg 6112	. . . . 5 ⊢ ((B ∈ NC ∧ A ∈ NC ) → (B ≤c A ↔ ∃d ∈ B ∃a ∈ A d ⊆ a))
3	2	ancoms 439	. . . 4 ⊢ ((A ∈ NC ∧ B ∈ NC ) → (B ≤c A ↔ ∃d ∈ B ∃a ∈ A d ⊆ a))
4	1, 3	anbi12d 691	. . 3 ⊢ ((A ∈ NC ∧ B ∈ NC ) → ((A ≤c B ∧ B ≤c 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))
6	5	2rexbii 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))
8	6, 7	bitri 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))
9	4, 8	syl6bbr 254	. 2 ⊢ ((A ∈ NC ∧ B ∈ NC ) → ((A ≤c B ∧ B ≤c A) ↔ ∃g ∈ A ∃d ∈ B ∃b ∈ B ∃a ∈ A (g ⊆ b ∧ d ⊆ a)))
10		ncseqnc 6128	. . . . . 6 ⊢ (A ∈ NC → (A = Nc g ↔ g ∈ A))
11		ncseqnc 6128	. . . . . 6 ⊢ (B ∈ NC → (B = Nc d ↔ d ∈ B))
12	10, 11	bi2anan9 843	. . . . 5 ⊢ ((A ∈ NC ∧ B ∈ NC ) → ((A = Nc g ∧ B = Nc d) ↔ (g ∈ A ∧ d ∈ B)))
13	12	biimpar 471	. . . 4 ⊢ (((A ∈ NC ∧ B ∈ NC ) ∧ (g ∈ A ∧ d ∈ B)) → (A = Nc g ∧ B = Nc d))
14		simplr 731	. . . . . . . . . . 11 ⊢ (((b ≈ d ∧ a ≈ g) ∧ (g ⊆ b ∧ d ⊆ a)) → a ≈ g)
15		ensym 6037	. . . . . . . . . . 11 ⊢ (a ≈ g ↔ g ≈ a)
16	14, 15	sylib 188	. . . . . . . . . 10 ⊢ (((b ≈ d ∧ a ≈ g) ∧ (g ⊆ b ∧ d ⊆ a)) → g ≈ a)
17		simprl 732	. . . . . . . . . . 11 ⊢ (((b ≈ d ∧ a ≈ g) ∧ (g ⊆ b ∧ d ⊆ a)) → g ⊆ b)
18		simpll 730	. . . . . . . . . . 11 ⊢ (((b ≈ d ∧ a ≈ g) ∧ (g ⊆ b ∧ d ⊆ a)) → b ≈ d)
19		simprr 733	. . . . . . . . . . 11 ⊢ (((b ≈ d ∧ a ≈ g) ∧ (g ⊆ b ∧ d ⊆ a)) → d ⊆ a)
20		sbthlem3 6205	. . . . . . . . . . 11 ⊢ (((a ≈ g ∧ g ⊆ b) ∧ (b ≈ d ∧ d ⊆ a)) → a ≈ b)
21	14, 17, 18, 19, 20	syl22anc 1183	. . . . . . . . . 10 ⊢ (((b ≈ d ∧ a ≈ g) ∧ (g ⊆ b ∧ d ⊆ a)) → a ≈ b)
22		entr 6038	. . . . . . . . . 10 ⊢ ((g ≈ a ∧ a ≈ b) → g ≈ b)
23	16, 21, 22	syl2anc 642	. . . . . . . . 9 ⊢ (((b ≈ d ∧ a ≈ g) ∧ (g ⊆ b ∧ d ⊆ a)) → g ≈ b)
24		entr 6038	. . . . . . . . 9 ⊢ ((g ≈ b ∧ b ≈ d) → g ≈ d)
25	23, 18, 24	syl2anc 642	. . . . . . . 8 ⊢ (((b ≈ d ∧ a ≈ g) ∧ (g ⊆ b ∧ d ⊆ a)) → g ≈ d)
26	25	ex 423	. . . . . . 7 ⊢ ((b ≈ d ∧ a ≈ g) → ((g ⊆ b ∧ d ⊆ a) → g ≈ d))
27		elnc 6125	. . . . . . . 8 ⊢ (b ∈ Nc d ↔ b ≈ d)
28		elnc 6125	. . . . . . . 8 ⊢ (a ∈ Nc g ↔ a ≈ g)
29	27, 28	anbi12i 678	. . . . . . 7 ⊢ ((b ∈ Nc d ∧ a ∈ Nc g) ↔ (b ≈ d ∧ a ≈ g))
30		vex 2862	. . . . . . . . 9 ⊢ g ∈ V
31	30	eqnc 6127	. . . . . . . 8 ⊢ ( Nc g = Nc d ↔ g ≈ d)
32	31	imbi2i 303	. . . . . . 7 ⊢ (((g ⊆ b ∧ d ⊆ a) → Nc g = Nc d) ↔ ((g ⊆ b ∧ d ⊆ a) → g ≈ d))
33	26, 29, 32	3imtr4i 257	. . . . . 6 ⊢ ((b ∈ Nc d ∧ a ∈ Nc g) → ((g ⊆ b ∧ d ⊆ a) → Nc g = Nc d))
34	33	rexlimivv 2743	. . . . 5 ⊢ (∃b ∈ Nc d∃a ∈ 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 d∃a ∈ 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)))
37	36	rexbidv 2635	. . . . . . 7 ⊢ (A = Nc g → (∃b ∈ Nc d∃a ∈ A (g ⊆ b ∧ d ⊆ a) ↔ ∃b ∈ Nc d∃a ∈ Nc g(g ⊆ b ∧ d ⊆ a)))
38	35, 37	sylan9bbr 681	. . . . . 6 ⊢ ((A = Nc g ∧ B = Nc d) → (∃b ∈ B ∃a ∈ A (g ⊆ b ∧ d ⊆ a) ↔ ∃b ∈ Nc d∃a ∈ Nc g(g ⊆ b ∧ d ⊆ a)))
39		eqeq12 2365	. . . . . 6 ⊢ ((A = Nc g ∧ B = Nc d) → (A = B ↔ Nc g = Nc d))
40	38, 39	imbi12d 311	. . . . 5 ⊢ ((A = Nc g ∧ B = Nc d) → ((∃b ∈ B ∃a ∈ A (g ⊆ b ∧ d ⊆ a) → A = B) ↔ (∃b ∈ Nc d∃a ∈ Nc g(g ⊆ b ∧ d ⊆ a) → Nc g = Nc d)))
41	34, 40	mpbiri 224	. . . 4 ⊢ ((A = Nc g ∧ B = Nc d) → (∃b ∈ B ∃a ∈ A (g ⊆ b ∧ d ⊆ a) → A = B))
42	13, 41	syl 15	. . 3 ⊢ (((A ∈ NC ∧ B ∈ NC ) ∧ (g ∈ A ∧ d ∈ B)) → (∃b ∈ B ∃a ∈ A (g ⊆ b ∧ d ⊆ a) → A = B))
43	42	rexlimdvva 2745	. 2 ⊢ ((A ∈ NC ∧ B ∈ NC ) → (∃g ∈ A ∃d ∈ B ∃b ∈ B ∃a ∈ A (g ⊆ b ∧ d ⊆ a) → A = B))
44	9, 43	sylbid 206	1 ⊢ ((A ∈ NC ∧ B ∈ NC ) → ((A ≤c B ∧ B ≤c A) → A = B))

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