Theorem cet 6235

Description: The exponent of a T-raising to a T-raising is always a cardinal. (Contributed by SF, 13-Mar-2015.)

Assertion

Ref	Expression
cet	⊢ ((M ∈ NC ∧ N ∈ NC ) → ( Tc M ↑c Tc N) ∈ NC )

Proof of Theorem cet

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tccl 6161	. . 3 ⊢ (M ∈ NC → Tc M ∈ NC )
2	1	adantr 451	. 2 ⊢ ((M ∈ NC ∧ N ∈ NC ) → Tc M ∈ NC )
3		tccl 6161	. . 3 ⊢ (N ∈ NC → Tc N ∈ NC )
4	3	adantl 452	. 2 ⊢ ((M ∈ NC ∧ N ∈ NC ) → Tc N ∈ NC )
5		elncs 6120	. . . 4 ⊢ (M ∈ NC ↔ ∃x M = Nc x)
6		tceq 6159	. . . . . . 7 ⊢ (M = Nc x → Tc M = Tc Nc x)
7	6	oveq1d 5538	. . . . . 6 ⊢ (M = Nc x → ( Tc M ↑c 0c) = ( Tc Nc x ↑c 0c))
8		vex 2863	. . . . . . . . . 10 ⊢ x ∈ V
9	8	ncelncsi 6122	. . . . . . . . 9 ⊢ Nc x ∈ NC
10		tccl 6161	. . . . . . . . 9 ⊢ ( Nc x ∈ NC → Tc Nc x ∈ NC )
11	9, 10	ax-mp 5	. . . . . . . 8 ⊢ Tc Nc x ∈ NC
12	8	pw1ex 4304	. . . . . . . . . 10 ⊢ ℘1x ∈ V
13	12	ncid 6124	. . . . . . . . 9 ⊢ ℘1x ∈ Nc ℘1x
14	8	tcnc 6226	. . . . . . . . 9 ⊢ Tc Nc x = Nc ℘1x
15	13, 14	eleqtrri 2426	. . . . . . . 8 ⊢ ℘1x ∈ Tc Nc x
16		ce0nnuli 6179	. . . . . . . 8 ⊢ (( Tc Nc x ∈ NC ∧ ℘1x ∈ Tc Nc x) → ( Tc Nc x ↑c 0c) ≠ ∅)
17	11, 15, 16	mp2an 653	. . . . . . 7 ⊢ ( Tc Nc x ↑c 0c) ≠ ∅
18		ce0nulnc 6185	. . . . . . . 8 ⊢ ( Tc Nc x ∈ NC → (( Tc Nc x ↑c 0c) ≠ ∅ ↔ ( Tc Nc x ↑c 0c) ∈ NC ))
19	11, 18	ax-mp 5	. . . . . . 7 ⊢ (( Tc Nc x ↑c 0c) ≠ ∅ ↔ ( Tc Nc x ↑c 0c) ∈ NC )
20	17, 19	mpbi 199	. . . . . 6 ⊢ ( Tc Nc x ↑c 0c) ∈ NC
21	7, 20	syl6eqel 2441	. . . . 5 ⊢ (M = Nc x → ( Tc M ↑c 0c) ∈ NC )
22	21	exlimiv 1634	. . . 4 ⊢ (∃x M = Nc x → ( Tc M ↑c 0c) ∈ NC )
23	5, 22	sylbi 187	. . 3 ⊢ (M ∈ NC → ( Tc M ↑c 0c) ∈ NC )
24	23	adantr 451	. 2 ⊢ ((M ∈ NC ∧ N ∈ NC ) → ( Tc M ↑c 0c) ∈ NC )
25		elncs 6120	. . . 4 ⊢ (N ∈ NC ↔ ∃x N = Nc x)
26		tceq 6159	. . . . . . 7 ⊢ (N = Nc x → Tc N = Tc Nc x)
27	26	oveq1d 5538	. . . . . 6 ⊢ (N = Nc x → ( Tc N ↑c 0c) = ( Tc Nc x ↑c 0c))
28	27, 20	syl6eqel 2441	. . . . 5 ⊢ (N = Nc x → ( Tc N ↑c 0c) ∈ NC )
29	28	exlimiv 1634	. . . 4 ⊢ (∃x N = Nc x → ( Tc N ↑c 0c) ∈ NC )
30	25, 29	sylbi 187	. . 3 ⊢ (N ∈ NC → ( Tc N ↑c 0c) ∈ NC )
31	30	adantl 452	. 2 ⊢ ((M ∈ NC ∧ N ∈ NC ) → ( Tc N ↑c 0c) ∈ NC )
32		cecl 6187	. 2 ⊢ ((( Tc M ∈ NC ∧ Tc N ∈ NC ) ∧ (( Tc M ↑c 0c) ∈ NC ∧ ( Tc N ↑c 0c) ∈ NC )) → ( Tc M ↑c Tc N) ∈ NC )
33	2, 4, 24, 31, 32	syl22anc 1183	1 ⊢ ((M ∈ NC ∧ N ∈ NC ) → ( Tc M ↑c Tc N) ∈ NC )

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∅c0 3551  ℘₁cpw1 4136  0_cc0c 4375  (class class class)co 5526   NC cncs 6089   Nc cnc 6092   T_c ctc 6094   ↑_c cce 6097

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 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-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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  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-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-compose 5749  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-map 6002  df-en 6030  df-ncs 6099  df-nc 6102  df-tc 6104  df-ce 6107

This theorem is referenced by:  ce2t  6236