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Mirrors > Home > NFE Home > Th. List > tceq | GIF version |
Description: Equality theorem for cardinal T operator. (Contributed by SF, 2-Mar-2015.) |
Ref | Expression |
---|---|
tceq | ⊢ (A = B → Tc A = Tc B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexeq 2808 | . . . 4 ⊢ (A = B → (∃y ∈ A x = Nc ℘1y ↔ ∃y ∈ B x = Nc ℘1y)) | |
2 | 1 | anbi2d 684 | . . 3 ⊢ (A = B → ((x ∈ NC ∧ ∃y ∈ A x = Nc ℘1y) ↔ (x ∈ NC ∧ ∃y ∈ B x = Nc ℘1y))) |
3 | 2 | iotabidv 4360 | . 2 ⊢ (A = B → (℩x(x ∈ NC ∧ ∃y ∈ A x = Nc ℘1y)) = (℩x(x ∈ NC ∧ ∃y ∈ B x = Nc ℘1y))) |
4 | df-tc 6103 | . 2 ⊢ Tc A = (℩x(x ∈ NC ∧ ∃y ∈ A x = Nc ℘1y)) | |
5 | df-tc 6103 | . 2 ⊢ Tc B = (℩x(x ∈ NC ∧ ∃y ∈ B x = Nc ℘1y)) | |
6 | 3, 4, 5 | 3eqtr4g 2410 | 1 ⊢ (A = B → Tc A = Tc B) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 ℘1cpw1 4135 ℩cio 4337 NC cncs 6088 Nc cnc 6091 Tc ctc 6093 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-rex 2620 df-uni 3892 df-iota 4339 df-tc 6103 |
This theorem is referenced by: tcdi 6164 tc2c 6166 tc11 6227 taddc 6228 tlecg 6229 letc 6230 ce0t 6231 ce2le 6232 cet 6233 tce2 6235 te0c 6236 ce0lenc1 6238 tlenc1c 6239 brtcfn 6245 nmembers1lem1 6267 nmembers1 6270 nchoicelem1 6287 nchoicelem2 6288 nchoicelem12 6298 nchoicelem16 6302 nchoicelem17 6303 nchoicelem19 6305 nchoice 6306 |
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