Theorem dfon4 36470

Description: Another quantifier-free definition of On. (Contributed by Scott Fenton, 4-May-2014.)

Assertion

Ref	Expression
dfon4	⊢ On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans ))

Proof of Theorem dfon4

Step	Hyp	Ref	Expression
1		dfon3 36469	. 2 ⊢ On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
2		df-ima 5668	. . . 4 ⊢ (( SSet ∖ ( I ∪ E )) “ Trans ) = ran (( SSet ∖ ( I ∪ E )) ↾ Trans )
3		df-res 5667	. . . . . 6 ⊢ (( SSet ∖ ( I ∪ E )) ↾ Trans ) = (( SSet ∖ ( I ∪ E )) ∩ ( Trans × V))
4		indif1 4228	. . . . . 6 ⊢ (( SSet ∖ ( I ∪ E )) ∩ ( Trans × V)) = (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))
5	3, 4	eqtri 2783	. . . . 5 ⊢ (( SSet ∖ ( I ∪ E )) ↾ Trans ) = (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))
6	5	rneqi 5921	. . . 4 ⊢ ran (( SSet ∖ ( I ∪ E )) ↾ Trans ) = ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))
7	2, 6	eqtri 2783	. . 3 ⊢ (( SSet ∖ ( I ∪ E )) “ Trans ) = ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))
8	7	difeq2i 4071	. 2 ⊢ (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans )) = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
9	1, 8	eqtr4i 2786	1 ⊢ On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans ))

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3450   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   I cid 5549   E cep 5554   × cxp 5653  ran crn 5656   ↾ cres 5657   “ cima 5658  Oncon0 6357   SSet csset 36409   Trans ctrans 36410

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539  df-fv 6541  df-1st 7986  df-2nd 7987  df-txp 36431  df-sset 36433  df-trans 36434

This theorem is used by: (None)