Theorem vfinnc 4472

Description: If the universe is finite, then there is a unique natural containing any set. Theorem X.1.22 of [Rosser] p. 527. (Contributed by SF, 19-Jan-2015.)

Assertion

Ref	Expression
vfinnc	⊢ ((A ∈ V ∧ V ∈ Fin ) → ∃!x ∈ Nn A ∈ x)

Distinct variable group:   x,A

Allowed substitution hint:   V(x)

Proof of Theorem vfinnc

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssv 3292	. . . 4 ⊢ A ⊆ V
2		ssfin 4471	. . . 4 ⊢ ((A ∈ V ∧ V ∈ Fin ∧ A ⊆ V) → A ∈ Fin )
3	1, 2	mp3an3 1266	. . 3 ⊢ ((A ∈ V ∧ V ∈ Fin ) → A ∈ Fin )
4		elfin 4421	. . 3 ⊢ (A ∈ Fin ↔ ∃x ∈ Nn A ∈ x)
5	3, 4	sylib 188	. 2 ⊢ ((A ∈ V ∧ V ∈ Fin ) → ∃x ∈ Nn A ∈ x)
6		nnceleq 4431	. . . . 5 ⊢ (((x ∈ Nn ∧ y ∈ Nn ) ∧ (A ∈ x ∧ A ∈ y)) → x = y)
7	6	ex 423	. . . 4 ⊢ ((x ∈ Nn ∧ y ∈ Nn ) → ((A ∈ x ∧ A ∈ y) → x = y))
8	7	rgen2a 2681	. . 3 ⊢ ∀x ∈ Nn ∀y ∈ Nn ((A ∈ x ∧ A ∈ y) → x = y)
9	8	a1i 10	. 2 ⊢ ((A ∈ V ∧ V ∈ Fin ) → ∀x ∈ Nn ∀y ∈ Nn ((A ∈ x ∧ A ∈ y) → x = y))
10		eleq2 2414	. . 3 ⊢ (x = y → (A ∈ x ↔ A ∈ y))
11	10	reu4 3031	. 2 ⊢ (∃!x ∈ Nn A ∈ x ↔ (∃x ∈ Nn A ∈ x ∧ ∀x ∈ Nn ∀y ∈ Nn ((A ∈ x ∧ A ∈ y) → x = y)))
12	5, 9, 11	sylanbrc 645	1 ⊢ ((A ∈ V ∧ V ∈ Fin ) → ∃!x ∈ Nn A ∈ x)

Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616  ∃!wreu 2617  Vcvv 2860   ⊆ wss 3258   Nn cnnc 4374   Fin cfin 4377

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-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-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-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-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381

This theorem is referenced by:  ncfinprop  4475