Theorem phi011 4599

Description: ( Phi A ∪ {0_c}) is one-to-one. Theorem X.2.4 of [Rosser] p. 282. (Contributed by SF, 3-Feb-2015.)

Assertion

Ref	Expression
phi011	⊢ (A = B ↔ ( Phi A ∪ {0c}) = ( Phi B ∪ {0c}))

Proof of Theorem phi011

Step	Hyp	Ref	Expression
1		phi11 4596	. 2 ⊢ (A = B ↔ Phi A = Phi B)
2		uneq1 3411	. . 3 ⊢ ( Phi A = Phi B → ( Phi A ∪ {0c}) = ( Phi B ∪ {0c}))
3		phi011lem1 4598	. . . 4 ⊢ (( Phi A ∪ {0c}) = ( Phi B ∪ {0c}) → Phi A ⊆ Phi B)
4		phi011lem1 4598	. . . . 5 ⊢ (( Phi B ∪ {0c}) = ( Phi A ∪ {0c}) → Phi B ⊆ Phi A)
5	4	eqcoms 2356	. . . 4 ⊢ (( Phi A ∪ {0c}) = ( Phi B ∪ {0c}) → Phi B ⊆ Phi A)
6	3, 5	eqssd 3289	. . 3 ⊢ (( Phi A ∪ {0c}) = ( Phi B ∪ {0c}) → Phi A = Phi B)
7	2, 6	impbii 180	. 2 ⊢ ( Phi A = Phi B ↔ ( Phi A ∪ {0c}) = ( Phi B ∪ {0c}))
8	1, 7	bitri 240	1 ⊢ (A = B ↔ ( Phi A ∪ {0c}) = ( Phi B ∪ {0c}))

Syntax hints:   ↔ wb 176   = wceq 1642   ∪ cun 3207   ⊆ wss 3257  {csn 3737  0_cc0c 4374   Phi cphi 4562

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

This theorem is referenced by:  proj2op  4601