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Theorem lec0cg 6198
 Description: Cardinal zero is a minimal element of cardinal less than or equal. Theorem XI.2.15 of [Rosser] p. 376. (Contributed by SF, 4-Mar-2015.)
Assertion
Ref Expression
lec0cg ((A V A) → 0cc A)

Proof of Theorem lec0cg
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ss 3579 . . . . . . 7 y
21jctr 526 . . . . . 6 (y A → (y A y))
32eximi 1576 . . . . 5 (y y Ay(y A y))
4 n0 3559 . . . . 5 (Ay y A)
5 df-rex 2620 . . . . 5 (y A yy(y A y))
63, 4, 53imtr4i 257 . . . 4 (Ay A y)
7 df-0c 4377 . . . . . 6 0c = {}
8 rexeq 2808 . . . . . 6 (0c = {} → (x 0c y A x yx {}y A x y))
97, 8ax-mp 5 . . . . 5 (x 0c y A x yx {}y A x y)
10 0ex 4110 . . . . . 6 V
11 sseq1 3292 . . . . . . 7 (x = → (x y y))
1211rexbidv 2635 . . . . . 6 (x = → (y A x yy A y))
1310, 12rexsn 3768 . . . . 5 (x {}y A x yy A y)
149, 13bitri 240 . . . 4 (x 0c y A x yy A y)
156, 14sylibr 203 . . 3 (Ax 0c y A x y)
1615adantl 452 . 2 ((A V A) → x 0c y A x y)
17 0cex 4392 . . . 4 0c V
18 brlecg 6112 . . . 4 ((0c V A V) → (0cc Ax 0c y A x y))
1917, 18mpan 651 . . 3 (A V → (0cc Ax 0c y A x y))
2019adantr 451 . 2 ((A V A) → (0cc Ax 0c y A x y))
2116, 20mpbird 223 1 ((A V A) → 0cc A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  Vcvv 2859   ⊆ wss 3257  ∅c0 3550  {csn 3737  0cc0c 4374   class class class wbr 4639   ≤c clec 6089 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-lec 6099 This theorem is referenced by:  le0nc  6200
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